Banach space

A Banach space is a complete normed space $(X,\; \backslash |\{\backslash cdot\}\backslash |).$

A normed space is a pairIt is common to read {{nowrap|"$X$ is a normed space"}} instead of the more technically correct but (usually) pedantic {{nowrap|"$(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is a normed space",}} especially if the norm is well known (for example, such as with $\backslash mathcal\{L\}^p$ spaces) or when there is no particular need to choose any one (equivalent) norm over any other (especially in the more abstract theory of topological vector spaces), in which case this norm (if needed) is often automatically assumed to be denoted by $\backslash |\{\backslash cdot\}\backslash |.$ However, in situations where emphasis is placed on the norm, it is common to see $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ written instead of $X.$ The technically correct definition of normed spaces as pairs $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ may also become important in the context of category theory where the distinction between the categories of normed spaces, normable spaces, metric spaces, TVSs, topological spaces, etc. is usually important.

$(X,\; \backslash |\{\backslash cdot\}\backslash |)$ consisting of a vector space $X$ over a scalar field $\backslash mathbb\{K\}$ (where $\backslash mathbb\{K\}$ is commonly $\backslash Reals$ or $\backslash Complex$) together with a distinguishedThis means that if the norm $\backslash |\{\backslash cdot\}\backslash |$ is replaced with a different norm $\backslash |\{\backslash cdot\}\backslash |\text{'}$ on $X,$ then $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is {{em|not}} the same normed space as $(X,\; \backslash |\{\backslash cdot\}\backslash |\text{'}),$ not even if the norms are equivalent. However, equivalence of norms on a given vector space does form an equivalence relation.

norm $\backslash |\{\backslash cdot\}\backslash |\; :\; X\; \backslash to\; \backslash Reals.$ Like all norms, this norm induces a translation invariantA metric $D$ on a vector space $X$ is said to be translation invariant if $D(x,\; y)\; =\; D(x\; +\; z,\; y\; +\; z)$ for all vectors $x,\; y,\; z\; \backslash in\; X.$ This happens if and only if $D(x,\; y)\; =\; D(x\; -\; y,\; 0)$ for all vectors $x,\; y\; \backslash in\; X.$ A metric that is induced by a norm is always translation invariant.

distance function, called the canonical or (norm) induced metric, defined for all vectors $x,\; y\; \backslash in\; X$ byBecause $\backslash |\{-z\}\backslash |\; =\; \backslash |z\backslash |$ for all $z\; \backslash in\; X,$ it is always true that $d(x,\; y)\; :=\; \backslash |y\; -\; x\backslash |\; =\; \backslash |x\; -\; y\backslash |$ for all $x,\; y\; \backslash in\; X.$ So the order of $x$ and $y$ in this definition does not matter.

$$d(x,\; y)\; :=\; \backslash |y\; -\; x\backslash |\; =\; \backslash |x\; -\; y\backslash |.$$

This makes $X$ into a metric space $(X,\; d).$

A sequence $x\_1,\; x\_2,\; \backslash ldots$ is called {{nobr|Cauchy in $(X,\; d)$}} or {{nowrap|$d$-Cauchy}} or {{nowrap|$\backslash |\{\backslash cdot\}\backslash |$-Cauchy}} if for every real $r\; >\; 0,$ there exists some index $N$ such that

whenever $m$ and $n$ are greater than $N.$

The normed space $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is called a Banach space and the canonical metric $d$ is called a complete metric if $(X,\; d)$ is a complete metric space, which by definition means for every Cauchy sequence $x\_1,\; x\_2,\; \backslash ldots$ in $(X,\; d),$ there exists some $x\; \backslash in\; X$ such that

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; x\_n\; =\; x\; \backslash ;\; \backslash text\{\; in\; \}\; (X,\; d),$$

where because $\backslash |x\_n\; -\; x\backslash |\; =\; d(x\_n,\; x),$ this sequence's convergence to $x$ can equivalently be expressed as

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash |x\_n\; -\; x\backslash |\; =\; 0\; \backslash ;\; \backslash text\{\; in\; \}\; \backslash Reals.$$

The norm $\backslash |\{\backslash cdot\}\backslash |$ of a normed space $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is called a {{visible anchor|complete norm|Complete norm}} if $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is a Banach space.

For any normed space $(X,\; \backslash |\{\backslash cdot\}\backslash |),$ there exists an L-semi-inner product $\backslash langle\backslash cdot,\; \backslash cdot\backslash rangle$ on $X$ such that $\backslash |x\backslash |\; =\; \backslash sqrt\{\backslash langle\; x,\; x\; \backslash rangle\}$ for all $x\; \backslash in\; X.${{cite journal |last1=Lumer |first1=G. |title=Semi-inner-product spaces |journal=Transactions of the American Mathematical Society |date=1961 |volume=100 |issue=1 |pages=29–43 |doi=10.1090/S0002-9947-1961-0133024-2|doi-access=free}} In general, there may be infinitely many L-semi-inner products that satisfy this condition and the proof of the existence of L-semi-inner products relies on the non-constructive Hahn–Banach theorem. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors.

A normed space $X$ is a Banach space if and only if each absolutely convergent series in $X$ converges to a value that lies within $X,$see Theorem 1.3.9, p. 20 in {{harvtxt|Megginson|1998}}. symbolically

The canonical metric $d$ of a normed space $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ induces the usual metric topology $\backslash tau\_d$ on $X,$ which is referred to as the canonical or norm induced topology.

Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise.

With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach.{{sfn|Wilansky|2013|p=29}} The norm $\backslash |\{\backslash cdot\}\backslash |\; :\; X\; \backslash to\; \backslash Reals$ is always a continuous function with respect to the topology that it induces.

The open and closed balls of radius $r\; >\; 0$ centered at a point $x\; \backslash in\; X$ are, respectively, the sets

Any such ball is a convex and bounded subset of $X,$ but a compact ball/neighborhood exists if and only if $X$ is finite-dimensional.

In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property.

If $x\_0$ is a vector and $s\; \backslash neq\; 0$ is a scalar, then

$$x\_0\; +\; s\backslash ,B\_r(x)\; =\; B\_\{|s|\; r\}(x\_0\; +\; s\; x)\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; x\_0\; +\; s\backslash ,C\_r(x)\; =\; C\_\{|s|\; r\}(x\_0\; +\; s\; x).$$

Using $s\; =\; 1$ shows that the norm-induced topology is translation invariant, which means that for any $x\; \backslash in\; X$ and $S\; \backslash subseteq\; X,$ the subset $S$ is open (respectively, closed) in $X$ if and only if its translation $x\; +\; S\; :=\; \backslash \{x\; +\; s\; \backslash mid\; s\; \backslash in\; S\backslash \}$ is open (respectively, closed).

Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include

$$\backslash \{B\_r(0)\; \backslash mid\; r\; >\; 0\backslash \},\; \backslash qquad\; \backslash \{C\_r(0)\; \backslash mid\; r\; >\; 0\backslash \},\; \backslash qquad\; \backslash \{B\_\{r\_n\}(0)\; \backslash mid\; n\; \backslash in\; \backslash N\backslash \},\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; \backslash \{C\_\{r\_n\}(0)\; \backslash mid\; n\; \backslash in\; \backslash N\backslash \},$$

where $r\_1,\; r\_2,\; \backslash ldots$ can be any sequence of positive real numbers that converges to $0$ in $\backslash R$ (common choices are $r\_n\; :=\; \backslash tfrac\{1\}\{n\}$ or $r\_n\; :=\; 1/2^n$).

So, for example, any open subset $U$ of $X$ can be written as a union

$$U\; =\; \backslash bigcup\_\{x\; \backslash in\; I\}\; B\_\{r\_x\}(x)\; =\; \backslash bigcup\_\{x\; \backslash in\; I\}\; x\; +\; B\_\{r\_x\}(0)\; =\; \backslash bigcup\_\{x\; \backslash in\; I\}\; x\; +\; r\_x\backslash ,B\_1(0)$$

indexed by some subset $I\; \backslash subseteq\; U,$ where each $r\_x$ may be chosen from the aforementioned sequence $r\_1,\; r\_2,\; \backslash ldots.$ (The open balls can also be replaced with closed balls, although the indexing set $I$ and radii $r\_x$ may then also need to be replaced).

Additionally, $I$ can always be chosen to be countable if $X$ is a {{em|separable space}}, which by definition means that $X$ contains some countable dense subset.

All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic.

Every separable infinite–dimensional Hilbert space is linearly isometrically isomorphic to the separable Hilbert sequence space $\backslash ell^2(\backslash N)$ with its usual norm $\backslash |\{\backslash cdot\}\backslash |\_2.$

The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space $\backslash prod\_\{i\; \backslash in\; \backslash N\}\; \backslash Reals$ of countably many copies of $\backslash Reals$ (this homeomorphism need not be a linear map).{{harvnb|Bessaga|Pełczyński|1975|p=189}}{{sfn|Anderson|Schori|1969|p=315}}

Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique up to a homeomorphism).

Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including $\backslash ell^2(\backslash N).$

In fact, $\backslash ell^2(\backslash N)$ is even homeomorphic to its own Unit sphere $\backslash \{x\; \backslash in\; \backslash ell^2(\backslash N)\; \backslash mid\; \backslash |x\backslash |\_2\; =\; 1\backslash \},$ which stands in sharp contrast to finite–dimensional spaces (the Euclidean plane $\backslash Reals^2$ is not homeomorphic to the unit circle, for instance).

This pattern in homeomorphism classes extends to generalizations of metrizable (locally Euclidean) topological manifolds known as {{em|metric Banach manifolds}}, which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space (metric Hilbert manifolds and metric Fréchet manifolds are defined similarly).{{sfn|Anderson|Schori|1969|p=315}}

For example, every open subset $U$ of a Banach space $X$ is canonically a metric Banach manifold modeled on $X$ since the inclusion map $U\; \backslash to\; X$ is an open local homeomorphism.

Using Hilbert space microbundles, David Henderson showed{{sfn|Henderson|1969|p=}} in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or Fréchet) space can be topologically embedded as an Open set of $\backslash ell^2(\backslash N)$ and, consequently, also admits a unique smooth structure making it into a $C^\backslash infty$ Hilbert manifold.

There is a compact subset $S$ of $\backslash ell^2(\backslash N)$ whose convex hull $\backslash operatorname\{co\}(S)$ is {{em|not}} closed and thus also {{em|not}} compact.Let $H$ be the separable Hilbert space $\backslash ell^2(\backslash N)$ of square-summable sequences with the usual norm $\backslash |\{\backslash cdot\}\backslash |\_2,$ and let $e\_n\; =\; (0,\; \backslash ldots,\; 0,\; 1,\; 0,\; \backslash ldots,\; 0)$ be the standard orthonormal basis (that is, each $e\_n$ has zeros in every position except for a $1$ in the $n$^th-position). The closed set $S\; =\; \backslash \{0\backslash \}\; \backslash cup\; \backslash \{\backslash tfrac\{1\}\{n\}\; e\_n\; \backslash mid\; n\; =\; 1,\; 2,\; \backslash ldots\backslash \}$ is compact (because it is sequentially compact) but its convex hull $\backslash operatorname\{co\}\; S$ is {{em|not}} a closed set because the point $h\; :=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash tfrac\{1\}\{2^n\}\; \backslash tfrac\{1\}\{n\}\; e\_n$ belongs to the closure of $\backslash operatorname\{co\}\; S$ in $H$ but $h\; \backslash not\backslash in\backslash operatorname\{co\}\; S$ (since every point $z=(z\_1,z\_2,\backslash ldots)\; \backslash in\; \backslash operatorname\{co\}\; S$ is a finite convex combination of elements of $S$ and so $z\_n\; =\; 0$ for all but finitely many coordinates, which is not true of $h$). However, like in all complete Hausdorff locally convex spaces, the {{em|closed}} convex hull $K\; :=\; \backslash overline\{\backslash operatorname\{co\}\}\; S$ of this compact subset is compact. The vector subspace $X\; :=\; \backslash operatorname\{span\}\; S\; =\; \backslash operatorname\{span\}\; \backslash \{e\_1,\; e\_2,\; \backslash ldots\backslash \}$ is a pre-Hilbert space when endowed with the substructure that the Hilbert space $H$ induces on it, but $X$ is not complete and $h\; \backslash not\backslash in\; C\; :=\; K\; \backslash cap\; X$ (since $h\; \backslash not\backslash in\; X$). The closed convex hull of $S$ in $X$ (here, "closed" means with respect to $X,$ and not to $H$ as before) is equal to $K\; \backslash cap\; X,$ which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of a compact subset might {{em|fail}} to be compact (although it will be precompact/totally bounded).{{sfn|Aliprantis|Border|2006|p=185}}

However, like in all Banach spaces, the Closed convex hull $\backslash overline\{\backslash operatorname\{co\}\}\; S$ of this (and every other) compact subset will be compact.{{sfn|Trèves|2006|p=145}} In a normed space that is not complete then it is in general {{em|not}} guaranteed that $\backslash overline\{\backslash operatorname\{co\}\}\; S$ will be compact whenever $S$ is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of $\backslash ell^2(\backslash N).$

This norm-induced topology also makes $(X,\; \backslash tau\_d)$ into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS $(X,\; \backslash tau\_d)$ is {{em|only}} a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is {{em|not}} associated with {{em|any}} particular norm or metric (both of which are "forgotten"). This Hausdorff TVS $(X,\; \backslash tau\_d)$ is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also {{em|normable}}, which by definition refers to any TVS whose topology is induced by some (possibly unknown) norm. Normable TVSs are characterized by being Hausdorff and having a bounded convex neighborhood of the origin.

All Banach spaces are barrelled spaces, which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the Banach–Steinhaus theorem holds.

The open mapping theorem implies that when $\backslash tau\_1$ and $\backslash tau\_2$ are topologies on $X$ that make both $(X,\; \backslash tau\_1)$ and $(X,\; \backslash tau\_2)$ into complete metrizable TVSes (for example, Banach or Fréchet spaces), if one topology is finer or coarser than the other, then they must be equal (that is, if $\backslash tau\_1\; \backslash subseteq\; \backslash tau\_2$ or $\backslash tau\_2\; \backslash subseteq\; \backslash tau\_1$ then $\backslash tau\_1\; =\; \backslash tau\_2$).{{sfn|Trèves|2006|pp=166–173}}

So, for example, if $(X,\; p)$ and $(X,\; q)$ are Banach spaces with topologies $\backslash tau\_p$ and $\backslash tau\_q,$ and if one of these spaces has some open ball that is also an open subset of the other space (or, equivalently, if one of $p\; :\; (X,\; \backslash tau\_q)\; \backslash to\; \backslash Reals$ or $q\; :\; (X,\; \backslash tau\_p)\; \backslash to\; \backslash Reals$ is continuous), then their topologies are identical and the norms $p$ and $q$ are equivalent.

Two norms, $p$ and $q,$ on a vector space $X$ are said to be equivalent if they induce the same topology;{{cite web|url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/equivnorms.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://kconrad.math.uconn.edu/blurbs/gradnumthy/equivnorms.pdf |archive-date=2022-10-09 |url-status=live|title=Equivalence of norms|last=Conrad|first=Keith|website=kconrad.math.uconn.edu|access-date=September 7, 2020}} this happens if and only if there exist real numbers $c,C\; >\; 0$ such that $c\backslash ,q(x)\; \backslash leq\; p(x)\; \backslash leq\; C\backslash ,q(x)$ for all $x\; \backslash in\; X.$ If $p$ and $q$ are two equivalent norms on a vector space $X$ then $(X,\; p)$ is a Banach space if and only if $(X,\; q)$ is a Banach space.

See this footnote for an example of a continuous norm on a Banach space that is {{em|not}} equivalent to that Banach space's given norm.Let $(C([0,\; 1]),\; |\{\backslash cdot\}\backslash |\_\{\backslash infty\})$ denote the Banach space of continuous functions with the supremum norm and let $\backslash tau\_\{\backslash infty\}$ denote the topology on $C([0,\; 1])$ induced by $\backslash |\{\backslash cdot\}\backslash |\_\{\backslash infty\}.$ The vector space $C([0,\; 1])$ can be identified (via the inclusion map) as a proper dense vector subspace $X$ of the $L^1$ space $(L^1([0,\; 1]),\; \backslash |\{\backslash cdot\}\backslash |\_1),$ which satisfies $\backslash |f\backslash |\_1\; \backslash leq\; \backslash |f\backslash |\_\{\backslash infty\}$ for all $f\; \backslash in\; X.$ Let $p$ denote the restriction of $\backslash |\{\backslash cdot\}\backslash |\_1$ to $X,$ which makes this map $p\; :\; X\; \backslash to\; \backslash R$ a norm on $X$ (in general, the restriction of any norm to any vector subspace will necessarily again be a norm). The normed space $(X,\; p)$ is {{em|not}} a Banach space since its completion is the proper superset $(L^1([0,\; 1]),\; \backslash |\{\backslash cdot\}\backslash |\_1).$ Because $p\; \backslash leq\; \backslash |\{\backslash cdot\}\backslash |\_\{\backslash infty\}$ holds on $X,$ the map $p\; :\; (X,\; \backslash tau\_\{\backslash infty\})\; \backslash to\; \backslash R$ is continuous. Despite this, the norm $p$ is {{em|not}} equivalent to the norm $\backslash |\{\backslash cdot\}\backslash |\_\{\backslash infty\}$ (because $(X,\; \backslash |\{\backslash cdot\}\backslash |\_\{\backslash infty\})$ is complete but $(X,\; p)$ is not).

All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.see Corollary 1.4.18, p. 32 in {{harvtxt|Megginson|1998}}.

A metric $D$ on a vector space $X$ is induced by a norm on $X$ if and only if $D$ is translation invariant and absolutely homogeneous, which means that $D(sx,\; sy)\; =\; |s|\; D(x,\; y)$ for all scalars $s$ and all $x,\; y\; \backslash in\; X,$ in which case the function $\backslash |x\backslash |\; :=\; D(x,\; 0)$ defines a norm on $X$ and the canonical metric induced by $\backslash |\{\backslash cdot\}\backslash |$ is equal to $D.$

Suppose that $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is a normed space and that $\backslash tau$ is the norm topology induced on $X.$ Suppose that $D$ is {{em|any}} metric on $X$ such that the topology that $D$ induces on $X$ is equal to $\backslash tau.$ If $D$ is translation invariant then $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is a Banach space if and only if $(X,\; D)$ is a complete metric space.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

If $D$ is {{em|not}} translation invariant, then it may be possible for $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ to be a Banach space but for $(X,\; D)$ to {{em|not}} be a complete metric space{{sfn|Narici|Beckenstein|2011|pp=47-51}} (see this footnoteThe normed space $(\backslash R,|\backslash cdot\; |)$ is a Banach space where the absolute value is a norm on the real line $\backslash R$ that induces the usual Euclidean topology on $\backslash R.$ Define a metric $D\; :\; \backslash R\; \backslash times\; \backslash R\; \backslash to\; \backslash R$ on $\backslash R$ by $D(x,\; y)\; =|\backslash arctan(x)\; -\; \backslash arctan(y)|$ for all $x,\; y\; \backslash in\; \backslash R.$ Just like {{nowrap|$|\backslash cdot|${{hsp}}'s}} induced metric, the metric $D$ also induces the usual Euclidean topology on $\backslash R.$ However, $D$ is not a complete metric because the sequence $x\_\{\backslash bull\}\; =\; (x\_i)\_\{i=1\}^\{\backslash infty\}$ defined by $x\_i\; :=\; i$ is a Cauchy sequence but it does not converge to any point of $\backslash R.$ As a consequence of not converging, this {{nowrap|$D$-Cauchy}} sequence cannot be a Cauchy sequence in $(\backslash R,|\backslash cdot\; |)$ (that is, it is not a Cauchy sequence with respect to the norm $|\backslash cdot|$) because if it was {{nowrap|$|\backslash cdot|$-Cauchy,}} then the fact that $(\backslash R,|\backslash cdot\; |)$ is a Banach space would imply that it converges (a contradiction).{{harvnb|Narici|Beckenstein|2011|pp=47–51}} for an example). In contrast, a theorem of Klee,{{sfn|Schaefer|Wolff|1999|p=35}}{{Cite journal|last1=Klee|first1=V. L.|title=Invariant metrics in groups (solution of a problem of Banach)|year=1952|journal=Proc. Amer. Math. Soc.|volume=3|issue=3|pages=484–487|url=https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf |archive-date=2022-10-09 |url-status=live|doi=10.1090/s0002-9939-1952-0047250-4|doi-access=free}}The statement of the theorem is: Let $d$ be {{em|any}} metric on a vector space $X$ such that the topology $\backslash tau$ induced by $d$ on $X$ makes $(X,\; \backslash tau)$ into a topological vector space. If $(X,\; d)$ is a complete metric space then $(X,\; \backslash tau)$ is a complete topological vector space. which also applies to all metrizable topological vector spaces, implies that if there exists {{em|any}}This metric $D$ is {{em|not}} assumed to be translation-invariant. So in particular, this metric $D$ does {{em|not}} even have to be induced by a norm. complete metric $D$ on $X$ that induces the norm topology $\backslash tau$ on $X,$ then $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ is a Banach space.

A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric.

Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences $\backslash R^\{\backslash N\}\; =\; \backslash prod\_\{i\; \backslash in\; \backslash N\}\; \backslash R$ with the product topology).

However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of norms.

It is even possible for a Fréchet space to have a topology that is induced by a countable family of {{em|norms}} (such norms would necessarily be continuous)A norm (or seminorm) $p$ on a topological vector space $(X,\; \backslash tau)$ is continuous if and only if the topology $\backslash tau\_p$ that $p$ induces on $X$ is coarser than $\backslash tau$ (meaning, $\backslash tau\_p\; \backslash subseteq\; \backslash tau$), which happens if and only if there exists some open ball $B$ in $(X,\; p)$ (such as maybe for example) that is open in $(X,\; \backslash tau).${{sfn|Trèves|2006|pp=57–69}}

but to not be a Banach/normable space because its topology can not be defined by any {{em|single}} norm.

An example of such a space is the Fréchet space $C^\{\backslash infty\}(K),$ whose definition can be found in the article on spaces of test functions and distributions.

There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces.

Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends {{em|only}} on vector subtraction and the topology $\backslash tau$ that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology $\backslash tau$ (and even applies to TVSs that are {{em|not}} even metrizable).

Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space.

If $(X,\; \backslash tau)$ is a metrizable topological vector space (such as any norm induced topology, for example), then $(X,\; \backslash tau)$ is a complete TVS if and only if it is a {{em|sequentially}} complete TVS, meaning that it is enough to check that every Cauchy {{em|sequence}} in $(X,\; \backslash tau)$ converges in $(X,\; \backslash tau)$ to some point of $X$ (that is, there is no need to consider the more general notion of arbitrary Cauchy nets).

If $(X,\; \backslash tau)$ is a topological vector space whose topology is induced by {{em|some}} (possibly unknown) norm (such spaces are called {{em|normable}}), then $(X,\; \backslash tau)$ is a complete topological vector space if and only if $X$ may be assigned a norm $\backslash |\{\backslash cdot\}\backslash |$ that induces on $X$ the topology $\backslash tau$ and also makes $(X,\; \backslash |\{\backslash cdot\}\backslash |)$ into a Banach space.

A Hausdorff locally convex topological vector space $X$ is normable if and only if its strong dual space $X\text{'}\_b$ is normable,{{sfn|Trèves|2006|p=201}} in which case $X\text{'}\_b$ is a Banach space ($X\text{'}\_b$ denotes the strong dual space of $X,$ whose topology is a generalization of the dual norm-induced topology on the continuous dual space $X\text{'}$; see this footnote$X\text{'}$ denotes the continuous dual space of $X.$ When $X\text{'}$ is endowed with the strong dual space topology, also called the topology of uniform convergence on bounded subsets of $X,$ then this is indicated by writing $X\text{'}\_b$ (sometimes, the subscript $\backslash beta$ is used instead of $b$). When $X$ is a normed space with norm $\backslash |\{\backslash cdot\}\backslash |$ then this topology is equal to the topology on $X\text{'}$ induced by the dual norm. In this way, the strong topology is a generalization of the usual dual norm-induced topology on $X\text{'}.$ for more details).

If $X$ is a metrizable locally convex TVS, then $X$ is normable if and only if $X\text{'}\_b$ is a Fréchet–Urysohn space.Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)

This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces.

Every normed space can be isometrically embedded onto a dense vector subspace of a Banach space, where this Banach space is called a completion of the normed space. This Hausdorff completion is unique up to isometric isomorphism.

More precisely, for every normed space $X,$ there exists a Banach space $Y$ and a mapping $T\; :\; X\; \backslash to\; Y$ such that $T$ is an isometric mapping and $T(X)$ is dense in $Y.$ If $Z$ is another Banach space such that there is an isometric isomorphism from $X$ onto a dense subset of $Z,$ then $Z$ is isometrically isomorphic to $Y.$

The Banach space $Y$ is the Hausdorff completion of the normed space $X.$ The underlying metric space for $Y$ is the same as the metric completion of $X,$ with the vector space operations extended from $X$ to $Y.$ The completion of $X$ is sometimes denoted by $\backslash widehat\{X\}.$

{{main|Bounded operator}}

If $X$ and $Y$ are normed spaces over the same ground field $\backslash mathbb\{K\},$ the set of all continuous $\backslash mathbb\{K\}$-linear maps $T\; :\; X\; \backslash to\; Y$ is denoted by $B(X,\; Y).$ In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space $X$ to another normed space is continuous if and only if it is bounded on the closed unit ball of $X.$ Thus, the vector space $B(X,\; Y)$ can be given the operator norm

$$\backslash |T\backslash |\; =\; \backslash sup\; \backslash \{\backslash |Tx\backslash |\_Y\; \backslash mid\; x\backslash in\; X,\backslash \; \backslash |x\backslash |\_X\; \backslash leq\; 1\backslash \}.$$

For $Y$ a Banach space, the space $B(X,\; Y)$ is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps; in that case the space $B(X,Y)$ reappears as a natural bifunctor.{{cite web|website=Annoying Precision|title=Banach spaces (and Lawvere metrics, and closed categories)|date=June 23, 2012|author=Qiaochu Yuan|url=https://qchu.wordpress.com/2012/06/23/banach-spaces-and-lawvere-metrics-and-closed-categories/}}

If $X$ is a Banach space, the space $B(X)\; =\; B(X,\; X)$ forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

If $X$ and $Y$ are normed spaces, they are isomorphic normed spaces if there exists a linear bijection $T\; :\; X\; \backslash to\; Y$ such that $T$ and its inverse $T^\{-1\}$ are continuous. If one of the two spaces $X$ or $Y$ is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces $X$ and $Y$ are isometrically isomorphic if in addition, $T$ is an isometry, that is, $\backslash |T(x)\backslash |\; =\; \backslash |x\backslash |$ for every $x$ in $X.$ The Banach–Mazur distance $d(X,\; Y)$ between two isomorphic but not isometric spaces $X$ and $Y$ gives a measure of how much the two spaces $X$ and $Y$ differ.

Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is $\backslash R$ or $\backslash Complex$) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.

If $f\; :\; X\; \backslash to\; \backslash R$ is a subadditive function (such as a norm, a sublinear function, or real linear functional), then{{sfn|Narici|Beckenstein|2011|pp=192-193}} $f$ is continuous at the origin if and only if $f$ is uniformly continuous on all of $X$; and if in addition $f(0)\; =\; 0$ then $f$ is continuous if and only if its absolute value $|f|\; :\; X\; \backslash to\; [0,\; \backslash infty)$ is continuous, which happens if and only if is an open subset of $X.${{sfn|Narici|Beckenstein|2011|pp=192-193}}The fact that being open implies that $f\; :\; X\; \backslash to\; \backslash R$ is continuous simplifies proving continuity because this means that it suffices to show that is open for $r\; :=\; 1$ and at $x\_0\; :=\; 0$ (where $f(0)\; =\; 0$) rather than showing this for {{em|all}} real $r\; >\; 0$ and {{em|all}} $x\_0\; \backslash in\; X.$

And very importantly for applying the Hahn–Banach theorem, a linear functional $f$ is continuous if and only if this is true of its real part $\backslash operatorname\{Re\}\; f$ and moreover, $\backslash |\backslash operatorname\{Re\}\; f\backslash |\; =\; \backslash |f\backslash |$ and the real part $\backslash operatorname\{Re\}\; f$ completely determines $f,$ which is why the Hahn–Banach theorem is often stated only for real linear functionals.

Also, a linear functional $f$ on $X$ is continuous if and only if the seminorm $|f|$ is continuous, which happens if and only if there exists a continuous seminorm $p\; :\; X\; \backslash to\; \backslash R$ such that $|f|\; \backslash leq\; p$; this last statement involving the linear functional $f$ and seminorm $p$ is encountered in many versions of the Hahn–Banach theorem.

The Cartesian product $X\; \backslash times\; Y$ of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used,{{harvtxt|Banach|1932|p=182}} such as

$$\backslash |(x,\; y)\backslash |\_1\; =\; \backslash |x\backslash |\; +\; \backslash |y\backslash |,\; \backslash qquad\; \backslash |(x,\; y)\backslash |\_\backslash infty\; =\; \backslash max(\backslash |x\backslash |,\; \backslash |y\backslash |)$$

which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product $X\; \backslash times\; Y$ (or the direct sum $X\; \backslash oplus\; Y$) is complete if and only if the two factors are complete.

If $M$ is a closed linear subspace of a normed space $X,$ there is a natural norm on the quotient space $X\; /\; M,$

$$\backslash |x\; +\; M\backslash |\; =\; \backslash inf\backslash limits\_\{m\; \backslash in\; M\}\; \backslash |x\; +\; m\backslash |.$$

The quotient $X\; /\; M$ is a Banach space when $X$ is complete.see pp. 17–19 in {{harvtxt|Carothers|2005}}. The quotient map from $X$ onto $X\; /\; M,$ sending $x\; \backslash in\; X$ to its class $x\; +\; M,$ is linear, onto, and of norm $1,$ except when $M\; =\; X,$ in which case the quotient is the null space.

The closed linear subspace $M$ of $X$ is said to be a complemented subspace of $X$ if $M$ is the range of a surjective bounded linear projection $P\; :\; X\; \backslash to\; M.$ In this case, the space $X$ is isomorphic to the direct sum of $M$ and $\backslash ker\; P,$ the kernel of the projection $P.$

Suppose that $X$ and $Y$ are Banach spaces and that $T\; \backslash in\; B(X,\; Y).$ There exists a canonical factorization of $T$ as

$$T\; =\; T\_1\; \backslash circ\; \backslash pi,\; \backslash quad\; T\; :\; X\; \backslash overset\{\backslash pi\}\{\{\}\backslash longrightarrow\{\}\}\; X/\backslash ker\; T\; \backslash overset\{T\_1\}\{\{\}\backslash longrightarrow\{\}\}\; Y$$

where the first map $\backslash pi$ is the quotient map, and the second map $T\_1$ sends every class $x\; +\; \backslash ker\; T$ in the quotient to the image $T(x)$ in $Y.$ This is well defined because all elements in the same class have the same image. The mapping $T\_1$ is a linear bijection from $X/\backslash ker\; T$ onto the range $T(X),$ whose inverse need not be bounded.

Basic examplessee {{harvtxt|Banach|1932}}, pp. 11-12. of Banach spaces include: the Lp spaces $L^p$ and their special cases, the sequence spaces $\backslash ell^p$ that consist of scalar sequences indexed by natural numbers $\backslash N$; among them, the space $\backslash ell^1$ of absolutely summable sequences and the space $\backslash ell^2$ of square summable sequences; the space $c\_0$ of sequences tending to zero and the space $\backslash ell^\{\backslash infty\}$ of bounded sequences; the space $C(K)$ of continuous scalar functions on a compact Hausdorff space $K,$ equipped with the max norm,

$$\backslash |f\backslash |\_\{C(K)\}\; =\; \backslash max\; \backslash \{\; |f(x)|\; \backslash mid\; x\; \backslash in\; K\; \backslash \},\; \backslash quad\; f\; \backslash in\; C(K).$$

According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some $C(K).$see {{harvtxt|Banach|1932}}, Th. 9 p. 185. For every separable Banach space $X,$ there is a closed subspace $M$ of $\backslash ell^1$ such that $X\; :=\; \backslash ell^1\; /\; M.$see Theorem 6.1, p. 55 in {{harvtxt|Carothers|2005}}

Any Hilbert space serves as an example of a Banach space. A Hilbert space $H$ on $\backslash mathbb\{K\}\; =\; \backslash Reals,\; \backslash Complex$ is complete for a norm of the form

$$\backslash |x\backslash |\_H\; =\; \backslash sqrt\{\backslash langle\; x,\; x\; \backslash rangle\},$$

where

$$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; :\; H\; \backslash times\; H\; \backslash to\; \backslash mathbb\{K\}$$

is the inner product, linear in its first argument that satisfies the following:

$$\backslash begin\{align\}$$

\langle y, x \rangle &= \overline{\langle x, y \rangle}, \quad \text{ for all } x, y \in H \\

\langle x, x \rangle & \geq 0, \quad \text{ for all } x \in H \\

\langle x,x \rangle = 0 \text{ if and only if } x &= 0.

\end{align}

For example, the space $L^2$ is a Hilbert space.

The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to $L^p$ spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.

A Banach algebra is a Banach space $A$ over $\backslash mathbb\{K\}\; =\; \backslash R$ or $\backslash Complex,$ together with a structure of algebra over $\backslash mathbb\{K\}$, such that the product map $A\; \backslash times\; A\; \backslash ni\; (a,\; b)\; \backslash mapsto\; ab\; \backslash in\; A$ is continuous. An equivalent norm on $A$ can be found so that $\backslash |ab\backslash |\; \backslash leq\; \backslash |a\backslash |\; \backslash |b\backslash |$ for all $a,\; b\; \backslash in\; A.$

• The Banach space $C(K)$ with the pointwise product, is a Banach algebra.
• The disk algebra $A(\backslash mathbf\{D\})$ consists of functions holomorphic in the open unit disk $D\; \backslash subseteq\; \backslash Complex$ and continuous on its closure: $\backslash overline\{\backslash mathbf\{D\}\}.$ Equipped with the max norm on $\backslash overline\{\backslash mathbf\{D\}\},$ the disk algebra $A(\backslash mathbf\{D\})$ is a closed subalgebra of $C\backslash left(\backslash overline\{\backslash mathbf\{D\}\}\backslash right).$
• The Wiener algebra $A(\backslash mathbf\{T\})$ is the algebra of functions on the unit circle $\backslash mathbf\{T\}$ with absolutely convergent Fourier series. Via the map associating a function on $\backslash mathbf\{T\}$ to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra $\backslash ell^1(Z),$ where the product is the convolution of sequences.
• For every Banach space $X,$ the space $B(X)$ of bounded linear operators on $X,$ with the composition of maps as product, is a Banach algebra.
• A C*-algebra is a complex Banach algebra $A$ with an antilinear involution $a\; \backslash mapsto\; a^*$ such that $\backslash |a^*\; a\backslash |\; =\; \backslash |a\backslash |^2.$ The space $B(H)$ of bounded linear operators on a Hilbert space $H$ is a fundamental example of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some $B(H).$ The space $C(K)$ of complex continuous functions on a compact Hausdorff space $K$ is an example of commutative C*-algebra, where the involution associates to every function $f$ its complex conjugate $\backslash overline\{f\}.$

{{main|Dual space}}

If $X$ is a normed space and $\backslash mathbb\{K\}$ the underlying field (either the reals or the complex numbers), the continuous dual space is the space of continuous linear maps from $X$ into $\backslash mathbb\{K\},$ or continuous linear functionals.

The notation for the continuous dual is $X\text{'}\; =\; B(X,\; \backslash mathbb\{K\})$ in this article.Several books about functional analysis use the notation $X^*$ for the continuous dual, for example {{harvtxt|Carothers|2005}}, {{harvtxt|Lindenstrauss|Tzafriri|1977}}, {{harvtxt|Megginson|1998}}, {{harvtxt|Ryan|2002}}, {{harvtxt|Wojtaszczyk|1991}}.

Since $\backslash mathbb\{K\}$ is a Banach space (using the absolute value as norm), the dual $X\text{'}$ is a Banach space, for every normed space $X.$ The Dixmier–Ng theorem characterizes the dual spaces of Banach spaces.

The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.

{{math theorem|name=Hahn–Banach theorem|math_statement=Let $X$ be a vector space over the field $\backslash mathbb\{K\}\; =\; \backslash R,\; \backslash Complex.$ Let further

• $Y\; \backslash subseteq\; X$ be a linear subspace,
• $p\; :\; X\; \backslash to\; \backslash R$ be a sublinear function and
• $f\; :\; Y\; \backslash to\; \backslash mathbb\{K\}$ be a linear functional so that $\backslash operatorname\{Re\}(f(y))\; \backslash leq\; p(y)$ for all $y\; \backslash in\; Y.$

Then, there exists a linear functional $F\; :\; X\; \backslash to\; \backslash mathbb\{K\}$ so that $$F\backslash big\backslash vert\_Y\; =\; f,\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X,\; \backslash \; \backslash \; \backslash operatorname\{Re\}(F(x))\; \backslash leq\; p(x).$$}}

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.Theorem 1.9.6, p. 75 in {{harvtxt|Megginson|1998}}

An important special case is the following: for every vector $x$ in a normed space $X,$ there exists a continuous linear functional $f$ on $X$ such that

$$f(x)\; =\; \backslash |x\backslash |\_X,\; \backslash quad\; \backslash |f\backslash |\_\{X\text{'}\}\; \backslash leq\; 1.$$

When $x$ is not equal to the $\backslash mathbf\{0\}$ vector, the functional $f$ must have norm one, and is called a norming functional for $x.$

The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane.

The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.see also Theorem 2.2.26, p. 179 in {{harvtxt|Megginson|1998}}

A subset $S$ in a Banach space $X$ is total if the linear span of $S$ is dense in $X.$ The subset $S$ is total in $X$ if and only if the only continuous linear functional that vanishes on $S$ is the $\backslash mathbf\{0\}$ functional: this equivalence follows from the Hahn–Banach theorem.

If $X$ is the direct sum of two closed linear subspaces $M$ and $N,$ then the dual $X\text{'}$ of $X$ is isomorphic to the direct sum of the duals of $M$ and $N.$see p. 19 in {{harvtxt|Carothers|2005}}.

If $M$ is a closed linear subspace in $X,$ one can associate the {{em|orthogonal of}} $M$ in the dual,

$$M^\{\backslash bot\}\; =\; \backslash \{\; x\text{'}\; \backslash in\; X\; \backslash mid\; x\text{'}(m)\; =\; 0\; \backslash text\{\; for\; all\; \}\; m\; \backslash in\; M\; \backslash \}.$$

The orthogonal $M^\{\backslash bot\}$ is a closed linear subspace of the dual. The dual of $M$ is isometrically isomorphic to $X\text{'}\; /\; M^\{\backslash bot\}.$

The dual of $X\; /\; M$ is isometrically isomorphic to $M^\{\backslash bot\}.$Theorems 1.10.16, 1.10.17 pp.94–95 in {{harvtxt|Megginson|1998}}

The dual of a separable Banach space need not be separable, but:

{{math theorem|name=TheoremTheorem 1.12.11, p. 112 in {{harvtxt|Megginson|1998}}|math_statement= Let $X$ be a normed space. If $X\text{'}$ is separable, then $X$ is separable.}}

When $X\text{'}$ is separable, the above criterion for totality can be used for proving the existence of a countable total subset in $X.$

The weak topology on a Banach space $X$ is the coarsest topology on $X$ for which all elements $x\text{'}$ in the continuous dual space $X\text{'}$ are continuous.

The norm topology is therefore finer than the weak topology.

It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.Theorem 2.5.16, p. 216 in {{harvtxt|Megginson|1998}}.

A norm-continuous linear map between two Banach spaces $X$ and $Y$ is also weakly continuous, that is, continuous from the weak topology of $X$ to that of $Y.$see II.A.8, p. 29 in {{harvtxt|Wojtaszczyk|1991}}

If $X$ is infinite-dimensional, there exist linear maps which are not continuous. The space $X^*$ of all linear maps from $X$ to the underlying field $\backslash mathbb\{K\}$ (this space $X^*$ is called the algebraic dual space, to distinguish it from $X\text{'}$ also induces a topology on $X$ which is finer than the weak topology, and much less used in functional analysis.

On a dual space $X\text{'},$ there is a topology weaker than the weak topology of $X\text{'},$ called the weak* topology.

It is the coarsest topology on $X\text{'}$ for which all evaluation maps $x\text{'}\; \backslash in\; X\text{'}\; \backslash mapsto\; x\text{'}(x),$ where $x$ ranges over $X,$ are continuous.

Its importance comes from the Banach–Alaoglu theorem.

{{math theorem|name=Banach–Alaoglu theorem|math_statement=Let $X$ be a normed vector space. Then the closed unit ball $B\; =\; \backslash \{x\; \backslash in\; X\; \backslash mid\; \backslash |x\backslash |\; \backslash leq\; 1\backslash \}$ of the dual space is compact in the weak* topology.}}

The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces.

When $X$ is separable, the unit ball $B\text{'}$ of the dual is a metrizable compact in the weak* topology.see Theorem 2.6.23, p. 231 in {{harvtxt|Megginson|1998}}.

The dual of $c\_0$ is isometrically isomorphic to $\backslash ell^1$: for every bounded linear functional $f$ on $c\_0,$ there is a unique element $y\; =\; \backslash \{y\_n\backslash \}\; \backslash in\; \backslash ell^1$ such that

$$f(x)\; =\; \backslash sum\_\{n\; \backslash in\; \backslash N\}\; x\_n\; y\_n,\; \backslash qquad\; x\; =\; \backslash \{x\_n\backslash \}\; \backslash in\; c\_0,\; \backslash \; \backslash \; \backslash text\{and\}\; \backslash \; \backslash \; \backslash |f\backslash |\_\{(c\_0)\text{'}\}\; =\; \backslash |y\backslash |\_\{\backslash ell\_1\}.$$

The dual of $\backslash ell^1$ is isometrically isomorphic to $\backslash ell^\{\backslash infty\}$.

The dual of Lebesgue space $L^p([0,\; 1])$ is isometrically isomorphic to $L^q([0,\; 1])$ when and $\backslash frac\{1\}\{p\}\; +\; \backslash frac\{1\}\{q\}\; =\; 1.$

For every vector $y$ in a Hilbert space $H,$ the mapping

$$x\; \backslash in\; H\; \backslash to\; f\_y(x)\; =\; \backslash langle\; x,\; y\; \backslash rangle$$

defines a continuous linear functional $f\_y$ on $H.$The Riesz representation theorem states that every continuous linear functional on $H$ is of the form $f\_y$ for a uniquely defined vector $y$ in $H.$

The mapping $y\; \backslash in\; H\; \backslash to\; f\_y$ is an antilinear isometric bijection from $H$ onto its dual $H\text{'}.$

When the scalars are real, this map is an isometric isomorphism.

When $K$ is a compact Hausdorff topological space, the dual $M(K)$ of $C(K)$ is the space of Radon measures in the sense of Bourbaki.see N. Bourbaki, (2004), "Integration I", Springer Verlag, {{ISBN|3-540-41129-1}}.

The subset $P(K)$ of $M(K)$ consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of $M(K).$

The extreme points of $P(K)$ are the Dirac measures on $K.$

The set of Dirac measures on $K,$ equipped with the w*-topology, is homeomorphic to $K.$

{{math theorem|name=Banach–Stone Theorem|math_statement=If $K$ and $L$ are compact Hausdorff spaces and if $C(K)$ and $C(L)$ are isometrically isomorphic, then the topological spaces $K$ and $L$ are homeomorphic.see also {{harvtxt|Banach|1932}}, p. 170 for metrizable $K$ and $L.$}}

The result has been extended by Amir{{cite journal |first=Dan |last=Amir |title=On isomorphisms of continuous function spaces |journal=Israel Journal of Mathematics |volume=3 |year=1965 |issue=4 |pages=205–210 |doi=10.1007/bf03008398 |doi-access=free |s2cid=122294213 }} and Cambern{{cite journal |first=M. |last=Cambern |title=A generalized Banach–Stone theorem |journal=Proc. Amer. Math. Soc. |volume=17 |year=1966 |issue=2 |pages=396–400 |doi=10.1090/s0002-9939-1966-0196471-9|doi-access=free}} And {{cite journal |first=M. |last=Cambern |title=On isomorphisms with small bound |journal=Proc. Amer. Math. Soc. |volume=18 |year=1967 |issue=6 |pages=1062–1066 |doi=10.1090/s0002-9939-1967-0217580-2|doi-access=free}} to the case when the multiplicative Banach–Mazur distance between $C(K)$ and $C(L)$ is

The theorem is no longer true when the distance is $=\; 2.${{cite journal |first=H. B. |last=Cohen |title=A bound-two isomorphism between $C(X)$ Banach spaces |journal=Proc. Amer. Math. Soc. |volume=50 |year=1975 |pages=215–217 |doi=10.1090/s0002-9939-1975-0380379-5|doi-access=free }}

In the commutative Banach algebra $C(K),$ the maximal ideals are precisely kernels of Dirac measures on $K,$

$$I\_x\; =\; \backslash ker\; \backslash delta\_x\; =\; \backslash \{f\; \backslash in\; C(K)\; \backslash mid\; f(x)\; =\; 0\backslash \},\; \backslash quad\; x\; \backslash in\; K.$$

More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology and the latter with the w*-topology.

In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual $A\text{'}.$

{{math theorem|math_statement= If $K$ is a compact Hausdorff space, then the maximal ideal space $\backslash Xi$ of the Banach algebra $C(K)$ is homeomorphic to $K.${{cite journal |last=Eilenberg |first=Samuel |title=Banach Space Methods in Topology |journal=Annals of Mathematics |date=1942 |volume=43 |issue=3 |pages=568–579 |doi=10.2307/1968812|jstor=1968812 }}}}

Not every unital commutative Banach algebra is of the form $C(K)$ for some compact Hausdorff space $K.$ However, this statement holds if one places $C(K)$ in the smaller category of commutative C*-algebras.

Gelfand's representation theorem for commutative C*-algebras states that every commutative unital C*-algebra $A$ is isometrically isomorphic to a $C(K)$ space.See for example {{cite book |first=W. |last=Arveson |year=1976 |title=An Invitation to C*-Algebra |publisher=Springer-Verlag |isbn=0-387-90176-0 }}

The Hausdorff compact space $K$ here is again the maximal ideal space, also called the spectrum of $A$ in the C*-algebra context.

{{See also|Bidual|Reflexive space|Semi-reflexive space}}

If $X$ is a normed space, the (continuous) dual $X\text{'}\text{'}$ of the dual $X\text{'}$ is called the {{visible anchor|bidual}} or {{visible anchor|second dual}} of $X.$

For every normed space $X,$ there is a natural map,

\|, \ \ x(f) = \lim_i f(x_i), \quad f \in X'.

The net may be replaced by a weakly*-convergent sequence when the dual $X\text{'}$ is separable.

On the other hand, elements of the bidual of $\backslash ell^1$ that are not in $\backslash ell^1$ cannot be weak*-limit of {{em|sequences}} in $\backslash ell^1,$ since $\backslash ell^1$ is weakly sequentially complete.

Here are the main general results about Banach spaces that go back to the time of Banach's book ({{harvtxt|Banach|1932}}) and are related to the Baire category theorem.

According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors.

Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.

{{math theorem|name=Banach–Steinhaus Theorem|math_statement=Let $X$ be a Banach space and $Y$ be a normed vector space. Suppose that $F$ is a collection of continuous linear operators from $X$ to $Y.$ The uniform boundedness principle states that if for all $x$ in $X$ we have then }}

The Banach–Steinhaus theorem is not limited to Banach spaces.

It can be extended for example to the case where $X$ is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood $U$ of $\backslash mathbf\{0\}$ in $X$ such that all $T$ in $F$ are uniformly bounded on $U,$

{{math theorem|name=The Open Mapping Theorem|math_statement=Let $X$ and $Y$ be Banach spaces and $T\; :\; X\; \backslash to\; Y$ be a surjective continuous linear operator, then $T$ is an open map.}}

{{math theorem|name=Corollary | math_statement = Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.}}

{{math theorem|name=The First Isomorphism Theorem for Banach spaces | math_statement= Suppose that $X$ and $Y$ are Banach spaces and that $T\; \backslash in\; B(X,\; Y).$ Suppose further that the range of $T$ is closed in $Y.$ Then $X\; /\; \backslash ker\; T$ is isomorphic to $T(X).$}}

This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.

{{math theorem|name=Corollary|math_statement=If a Banach space $X$ is the internal direct sum of closed subspaces $M\_1,\; \backslash ldots,\; M\_n,$ then $X$ is isomorphic to $M\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; M\_n.$}}

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from $M\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; M\_n$ onto $X$ sending $m\_1,\; \backslash cdots,\; m\_n$ to the sum $m\_1\; +\; \backslash cdots\; +\; m\_n.$

{{math theorem|name=The Closed Graph Theorem|math_statement= Let $T\; :\; X\; \backslash to\; Y$ be a linear mapping between Banach spaces. The graph of $T$ is closed in $X\; \backslash times\; Y$ if and only if $T$ is continuous.}}

{{main|Reflexive space}}

The normed space $X$ is called reflexive when the natural map

is surjective. Reflexive normed spaces are Banach spaces.

{{math theorem| math_statement = If $X$ is a reflexive Banach space, every closed subspace of $X$ and every quotient space of $X$ are reflexive.}}

This is a consequence of the Hahn–Banach theorem.

Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space $X$ onto the Banach space $Y,$ then $Y$ is reflexive.

{{math theorem| math_statement = If $X$ is a Banach space, then $X$ is reflexive if and only if $X\text{'}$ is reflexive.}}

{{math theorem|name=Corollary | math_statement = Let $X$ be a reflexive Banach space. Then $X$ is separable if and only if $X\text{'}$ is separable.}}

Indeed, if the dual $Y\text{'}$ of a Banach space $Y$ is separable, then $Y$ is separable.

If $X$ is reflexive and separable, then the dual of $X\text{'}$ is separable, so $X\text{'}$ is separable.

{{math theorem| math_statement = Suppose that $X\_1,\; \backslash ldots,\; X\_n$ are normed spaces and that $X\; =\; X\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; X\_n.$ Then $X$ is reflexive if and only if each $X\_j$ is reflexive.}}

Hilbert spaces are reflexive. The $L^p$ spaces are reflexive when More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem.

The spaces $c\_0,\; \backslash ell^1,\; L^1([0,\; 1]),\; C([0,\; 1])$ are not reflexive.

In these examples of non-reflexive spaces $X,$ the bidual $X\text{'}\text{'}$ is "much larger" than $X.$

Namely, under the natural isometric embedding of $X$ into $X$ given by the Hahn–Banach theorem, the quotient $X$ / X is infinite-dimensional, and even nonseparable.

However, Robert C. James has constructed an example{{cite journal|author = R. C. James|title=A non-reflexive Banach space isometric with its second conjugate space|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=37|pages=174–177|year=1951|issue=3 | doi=10.1073/pnas.37.3.174 | pmc=1063327|pmid=16588998|bibcode=1951PNAS...37..174J |doi-access=free}} of a non-reflexive space, usually called "the James space" and denoted by $J,$see {{harvtxt|Lindenstrauss|Tzafriri|1977}}, p. 25. such that the quotient $J\text{'}\text{'}\; /\; J$ is one-dimensional.

Furthermore, this space $J$ is isometrically isomorphic to its bidual.

{{math theorem| math_statement = A Banach space $X$ is reflexive if and only if its unit ball is compact in the weak topology.}}

When $X$ is reflexive, it follows that all closed and bounded convex subsets of $X$ are weakly compact.

In a Hilbert space $H,$ the weak compactness of the unit ball is very often used in the following way: every bounded sequence in $H$ has weakly convergent subsequences.

Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.

For example, every convex continuous function on the unit ball $B$ of a reflexive space attains its minimum at some point in $B.$

As a special case of the preceding result, when $X$ is a reflexive space over $\backslash R,$ every continuous linear functional $f$ in $X\text{'}$ attains its maximum $\backslash |f\backslash |$ on the unit ball of $X.$

The following theorem of Robert C. James provides a converse statement.

{{math theorem| name = James' Theorem | math_statement = For a Banach space the following two properties are equivalent:

• $X$ is reflexive.
• for all $f$ in $X\text{'}$ there exists $x\; \backslash in\; X$ with $\backslash |x\backslash |\; \backslash leq\; 1,$ so that $f(x)\; =\; \backslash |f\backslash |.$}}

The theorem can be extended to give a characterization of weakly compact convex sets.

On every non-reflexive Banach space $X,$ there exist continuous linear functionals that are not norm-attaining.

However, the Bishop–Phelps theorem{{cite journal|last1=bishop|first1=See E.|last2=Phelps|first2=R.|year=1961|title=A proof that every Banach space is subreflexive|journal=Bull. Amer. Math. Soc.|volume=67|pages=97–98|doi=10.1090/s0002-9904-1961-10514-4|doi-access=free }} states that norm-attaining functionals are norm dense in the dual $X\text{'}$ of $X.$

A sequence $\backslash \{x\_n\backslash \}$ in a Banach space $X$ is weakly convergent to a vector $x\; \backslash in\; X$ if $\backslash \{f(x\_n)\backslash \}$ converges to $f(x)$ for every continuous linear functional $f$ in the dual $X\text{'}.$ The sequence $\backslash \{x\_n\backslash \}$ is a weakly Cauchy sequence if $\backslash \{f(x\_n)\backslash \}$ converges to a scalar limit $L(f)$ for every $f$ in $X\text{'}.$

A sequence $\backslash \{f\_n\backslash \}$ in the dual $X\text{'}$ is weakly* convergent to a functional $f\; \backslash in\; X\text{'}$ if $f\_n(x)$ converges to $f(x)$ for every $x$ in $X.$

Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.

When the sequence $\backslash \{x\_n\backslash \}$ in $X$ is a weakly Cauchy sequence, the limit $L$ above defines a bounded linear functional on the dual $X\text{'},$ that is, an element $L$ of the bidual of $X,$ and $L$ is the limit of $\backslash \{x\_n\backslash \}$ in the weak*-topology of the bidual.

The Banach space $X$ is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in $X.$

It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.

{{math theorem| name = Theorem see III.C.14, p. 140 in {{harvtxt|Wojtaszczyk|1991}}. | math_statement = For every measure $\backslash mu,$ the space $L^1(\backslash mu)$ is weakly sequentially complete.}}

An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the $\backslash mathbf\{0\}$ vector.

The unit vector basis of $\backslash ell^p$ for or of $c\_0,$ is another example of a weakly null sequence, that is, a sequence that converges weakly to $\backslash mathbf\{0\}.$

For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to $\backslash mathbf\{0\}.$see Corollary 2, p. 11 in {{harvtxt|Diestel|1984}}.

The unit vector basis of $\backslash ell^1$ is not weakly Cauchy.

Weakly Cauchy sequences in $\backslash ell^1$ are weakly convergent, since $L^1$-spaces are weakly sequentially complete.

Actually, weakly convergent sequences in $\backslash ell^1$ are norm convergent.see p. 85 in {{harvtxt|Diestel|1984}}. This means that $\backslash ell^1$ satisfies Schur's property.

Weakly Cauchy sequences and the $\backslash ell^1$ basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.{{cite journal|last1=Rosenthal|first1=Haskell P|year=1974|title=A characterization of Banach spaces containing ℓ¹|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=71|issue=6| pages=2411–2413 | doi=10.1073/pnas.71.6.2411|pmid=16592162|pmc=388466|arxiv=math.FA/9210205|bibcode=1974PNAS...71.2411R|doi-access=free}} Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in {{cite journal| last1=Dor|first1=Leonard E|year=1975|title=On sequences spanning a complex ℓ¹ space|journal=Proc. Amer. Math. Soc. | volume=47|pages=515–516|doi=10.1090/s0002-9939-1975-0358308-x|doi-access=free}}

{{math theorem| name = Theoremsee p. 201 in {{harvtxt|Diestel|1984}}. | math_statement = Let $\backslash \{x\_n\backslash \}\_\{n\; \backslash in\; \backslash N\}$ be a bounded sequence in a Banach space. Either $\backslash \{x\_n\backslash \}\_\{n\; \backslash in\; \backslash N\}$ has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of $\backslash ell^1.$}}

A complement to this result is due to Odell and Rosenthal (1975).

{{math theorem| name = Theorem{{citation|last1=Odell|first1=Edward W.|last2=Rosenthal|first2=Haskell P.|title=A double-dual characterization of separable Banach spaces containing ℓ¹|journal=Israel Journal of Mathematics|volume=20|year=1975|issue=3–4 |pages=375–384|doi=10.1007/bf02760341|doi-access=free|s2cid=122391702|url=http://dml.cz/bitstream/handle/10338.dmlcz/133414/CommentatMathUnivCarolRetro_50-2009-1_5.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://dml.cz/bitstream/handle/10338.dmlcz/133414/CommentatMathUnivCarolRetro_50-2009-1_5.pdf |archive-date=2022-10-09 |url-status=live}}. | math_statement = Let $X$ be a separable Banach space. The following are equivalent:

• The space $X$ does not contain a closed subspace isomorphic to $\backslash ell^1.$
• Every element of the bidual $X\text{'}\text{'}$ is the weak*-limit of a sequence $\backslash \{x\_n\backslash \}$ in $X.$}}

By the Goldstine theorem, every element of the unit ball $B$ of $X$ is weak*-limit of a net in the unit ball of $X.$ When $X$ does not contain $\backslash ell^1,$ every element of $B\text{'}\text{'}$ is weak*-limit of a {{em|sequence}} in the unit ball of $X.$Odell and Rosenthal, Sublemma p. 378 and Remark p. 379.

When the Banach space $X$ is separable, the unit ball of the dual $X\text{'},$ equipped with the weak*-topology, is a metrizable compact space $K,$ and every element $x$ in the bidual $X$ defines a bounded function on $K$:

$$x\text{'}\; \backslash in\; K\; \backslash mapsto\; x$$(x'), \quad |x(x')| \leq \|x''\|.

This function is continuous for the compact topology of $K$ if and only if $x$ is actually in $X,$ considered as subset of $X$.

Assume in addition for the rest of the paragraph that $X$ does not contain $\backslash ell^1.$

By the preceding result of Odell and Rosenthal, the function $x\text{'}\text{'}$ is the pointwise limit on $K$ of a sequence $\backslash \{x\_n\backslash \}\; \backslash subseteq\; X$ of continuous functions on $K,$ it is therefore a first Baire class function on $K.$

The unit ball of the bidual is a pointwise compact subset of the first Baire class on $K.$for more on pointwise compact subsets of the Baire class, see {{citation|last1=Bourgain|first1=Jean|author1-link=Jean Bourgain|last2=Fremlin|first2=D. H.|last3=Talagrand |first3=Michel|title=Pointwise Compact Sets of Baire-Measurable Functions|journal=Am. J. Math.|volume=100|year=1978|issue=4|pages=845–886|jstor=2373913|doi=10.2307/2373913}}.

When $X$ is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences.

This applies to separable reflexive spaces, but more is true in this case, as stated below.

The weak topology of a Banach space $X$ is metrizable if and only if $X$ is finite-dimensional.see Proposition 2.5.14, p. 215 in {{harvtxt|Megginson|1998}}. If the dual $X\text{'}$ is separable, the weak topology of the unit ball of $X$ is metrizable.

This applies in particular to separable reflexive Banach spaces.

Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.

{{math theorem| name = Eberlein–Šmulian theoremsee for example p. 49, II.C.3 in {{harvtxt|Wojtaszczyk|1991}}. | math_statement = A set $A$ in a Banach space is relatively weakly compact if and only if every sequence $\backslash \{a\_n\backslash \}$ in $A$ has a weakly convergent subsequence.}}

A Banach space $X$ is reflexive if and only if each bounded sequence in $X$ has a weakly convergent subsequence.see Corollary 2.8.9, p. 251 in {{harvtxt|Megginson|1998}}.

A weakly compact subset $A$ in $\backslash ell^1$ is norm-compact. Indeed, every sequence in $A$ has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of $\backslash ell^1.$

{{main|Type and cotype of a Banach space}}

A way to classify Banach spaces is through the probabilistic notion of type and cotype, these two measure how far a Banach space is from a Hilbert space.

{{main|Schauder basis}}

A Schauder basis in a Banach space $X$ is a sequence $\backslash \{e\_n\backslash \}\_\{n\; \backslash geq\; 0\}$ of vectors in $X$ with the property that for every vector $x\; \backslash in\; X,$ there exist {{em|uniquely}} defined scalars $\backslash \{x\_n\backslash \}\_\{n\; \backslash geq\; 0\}$ depending on $x,$ such that

$$x\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x\_n\; e\_n,\; \backslash quad\; \backslash textit\{i.e.,\}\; \backslash quad\; x\; =\; \backslash lim\_n\; P\_n(x),\; \backslash \; P\_n(x)\; :=\; \backslash sum\_\{k=0\}^n\; x\_k\; e\_k.$$

Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.

It follows from the Banach–Steinhaus theorem that the linear mappings $\backslash \{P\_n\backslash \}$ are uniformly bounded by some constant $C.$

Let $\backslash \{e\_n^*\backslash \}$ denote the coordinate functionals which assign to every $x$ in $X$ the coordinate $x\_n$ of $x$ in the above expansion.

They are called biorthogonal functionals. When the basis vectors have norm $1,$ the coordinate functionals $\backslash \{e\_n^*\backslash \}$ have norm $\{\}\backslash leq\; 2\; C$ in the dual of $X.$

Most classical separable spaces have explicit bases.

The Haar system $\backslash \{h\_n\backslash \}$ is a basis for $L^p([0,\; 1])$ when

The trigonometric system is a basis in $L^p(\backslash mathbf\{T\})$ when

The Schauder system is a basis in the space $C([0,\; 1]).$see {{harvtxt|Lindenstrauss|Tzafriri|1977}} p. 3.

The question of whether the disk algebra $A(\backslash mathbf\{D\})$ has a basisthe question appears p. 238, §3 in Banach's book, {{harvtxt|Banach|1932}}. remained open for more than forty years, until Bočkarev showed in 1974 that $A(\backslash mathbf\{D\})$ admits a basis constructed from the Franklin system.see S. V. Bočkarev, "Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system". (Russian) Mat. Sb. (N.S.) 95(137) (1974), 3–18, 159.

Since every vector $x$ in a Banach space $X$ with a basis is the limit of $P\_n(x),$ with $P\_n$ of finite rank and uniformly bounded, the space $X$ satisfies the bounded approximation property.

The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.see {{cite journal|last1=Enflo|first1=P.|year=1973|title=A counterexample to the approximation property in Banach spaces|journal=Acta Math.|volume=130|pages=309–317| doi=10.1007/bf02392270| s2cid=120530273 | doi-access=free}}

Robert C. James characterized reflexivity in Banach spaces with a basis: the space $X$ with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.see R.C. James, "Bases and reflexivity of Banach spaces". Ann. of Math. (2) 52, (1950). 518–527. See also {{harvtxt|Lindenstrauss|Tzafriri|1977}} p. 9.

In this case, the biorthogonal functionals form a basis of the dual of $X.$

{{main|Tensor product|Topological tensor product}}

thumb

Let $X$ and $Y$ be two $\backslash mathbb\{K\}$-vector spaces. The tensor product $X\; \backslash otimes\; Y$ of $X$ and $Y$ is a $\backslash mathbb\{K\}$-vector space $Z$ with a bilinear mapping $T\; :\; X\; \backslash times\; Y\; \backslash to\; Z$ which has the following universal property:

:If $T\_1\; :\; X\; \backslash times\; Y\; \backslash to\; Z\_1$ is any bilinear mapping into a $\backslash mathbb\{K\}$-vector space $Z\_1,$ then there exists a unique linear mapping $f\; :\; Z\; \backslash to\; Z\_1$ such that $T\_1\; =\; f\; \backslash circ\; T.$

The image under $T$ of a couple $(x,\; y)$ in $X\; \backslash times\; Y$ is denoted by $x\; \backslash otimes\; y,$ and called a simple tensor.

Every element $z$ in $X\; \backslash otimes\; Y$ is a finite sum of such simple tensors.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.see A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires". Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp., and A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques". Bol. Soc. Mat. São Paulo 8 1953 1–79.

In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor productsee chap. 2, p. 15 in {{harvtxt|Ryan|2002}}. of two Banach spaces $X$ and $Y$ is the {{em|completion}} $X\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; Y$ of the algebraic tensor product $X\; \backslash otimes\; Y$ equipped with the projective tensor norm, and similarly for the injective tensor productsee chap. 3, p. 45 in {{harvtxt|Ryan|2002}}. $X\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; Y.$

Grothendieck proved in particular thatsee Example. 2.19, p. 29, and pp. 49–50 in {{harvtxt|Ryan|2002}}.

$$\backslash begin\{align\}$$

C(K) \widehat{\otimes}_\varepsilon Y &\simeq C(K, Y), \\

L^1([0, 1]) \widehat{\otimes}_\pi Y &\simeq L^1([0, 1], Y),

\end{align}

where $K$ is a compact Hausdorff space, $C(K,\; Y)$ the Banach space of continuous functions from $K$ to $Y$ and $L^1([0,\; 1],\; Y)$ the space of Bochner-measurable and integrable functions from $[0,\; 1]$ to $Y,$ and where the isomorphisms are isometric.

The two isomorphisms above are the respective extensions of the map sending the tensor $f\; \backslash otimes\; y$ to the vector-valued function $s\; \backslash in\; K\; \backslash to\; f(s)\; y\; \backslash in\; Y.$

Let $X$ be a Banach space. The tensor product $X\text{'}\; \backslash widehat\; \backslash otimes\_\backslash varepsilon\; X$ is identified isometrically with the closure in $B(X)$ of the set of finite rank operators.

When $X$ has the approximation property, this closure coincides with the space of compact operators on $X.$

For every Banach space $Y,$ there is a natural norm $1$ linear map

$$Y\; \backslash widehat\backslash otimes\_\backslash pi\; X\; \backslash to\; Y\; \backslash widehat\backslash otimes\_\backslash varepsilon\; X$$

obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when $Y$ is the dual of $X.$

Precisely, for every Banach space $X,$ the map

$$X\text{'}\; \backslash widehat\; \backslash otimes\_\backslash pi\; X\; \backslash \; \backslash longrightarrow\; X\text{'}\; \backslash widehat\; \backslash otimes\_\backslash varepsilon\; X$$

is one-to-one if and only if $X$ has the approximation property.see Proposition 4.6, p. 74 in {{harvtxt|Ryan|2002}}.

Grothendieck conjectured that $X\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; Y$ and $X\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; Y$ must be different whenever $X$ and $Y$ are infinite-dimensional Banach spaces.

This was disproved by Gilles Pisier in 1983.see Pisier, Gilles (1983), "Counterexamples to a conjecture of Grothendieck", Acta Math. 151:181–208.

Pisier constructed an infinite-dimensional Banach space $X$ such that $X\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; X$ and $X\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; X$ are equal. Furthermore, just as Enflo's example, this space $X$ is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space $B(\backslash ell^2)$ does not have the approximation property.see Szankowski, Andrzej (1981), "$B(H)$ does not have the approximation property", Acta Math. 147: 89–108. Ryan claims that this result is due to Per Enflo, p. 74 in {{harvtxt|Ryan|2002}}.

A necessary and sufficient condition for the norm of a Banach space $X$ to be associated to an inner product is the parallelogram identity:

{{math theorem| name = Parallelogram identity | math_statement = for all $x,\; y\; \backslash in\; X\; :\; \backslash qquad\; \backslash |x+y\backslash |^2\; +\; \backslash |x-y\backslash |^2\; =\; 2(\backslash |x\backslash |^2\; +\; \backslash |y\backslash |^2).$}}

It follows, for example, that the Lebesgue space $L^p([0,\; 1])$ is a Hilbert space only when $p\; =\; 2.$

If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:

$$\backslash langle\; x,\; y\backslash rangle\; =\; \backslash tfrac\{1\}\{4\}(\backslash |x+y\backslash |^2\; -\; \backslash |x-y\backslash |^2).$$

For complex scalars, defining the inner product so as to be $\backslash Complex$-linear in $x,$ antilinear in $y,$ the polarization identity gives:

$$\backslash langle\; x,y\backslash rangle\; =\; \backslash tfrac\{1\}\{4\}\backslash left(\backslash |x+y\backslash |^2\; -\; \backslash |x-y\backslash |^2\; +\; i(\backslash |x+iy\backslash |^2\; -\; \backslash |x-iy\backslash |^2)\backslash right).$$

To see that the parallelogram law is sufficient, one observes in the real case that $\backslash langle\; x,\; y\; \backslash rangle$ is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and $\backslash langle\; i\; x,\; y\; \backslash rangle\; =\; i\; \backslash langle\; x,\; y\; \backslash rangle.$ The parallelogram law implies that $\backslash langle\; x,\; y\; \backslash rangle$ is additive in $x.$

It follows that it is linear over the rationals, thus linear by continuity.

Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available.

The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant $c\; \backslash geq\; 1$: Kwapień proved that if

$$c^\{-2\}\; \backslash sum\_\{k=1\}^n\; \backslash |x\_k\backslash |^2\; \backslash leq\; \backslash operatorname\{Ave\}\_\{\backslash pm\}\; \backslash left\backslash |\backslash sum\_\{k=1\}^n\; \backslash pm\; x\_k\backslash right\backslash |^2\; \backslash leq\; c^2\; \backslash sum\_\{k=1\}^n\; \backslash |x\_k\backslash |^2$$

for every integer $n$ and all families of vectors $\backslash \{x\_1,\; \backslash ldots,\; x\_n\backslash \}\; \backslash subseteq\; X,$ then the Banach space $X$ is isomorphic to a Hilbert space.see Kwapień, S. (1970), "A linear topological characterization of inner-product spaces", Studia Math. 38:277–278.

Here, $\backslash operatorname\{Ave\}\_\{\backslash pm\}$ denotes the average over the $2^n$ possible choices of signs $\backslash pm\; 1.$

In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.

Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space.{{cite journal

|last1=Lindenstrauss|first1=Joram

|last2=Tzafriri|first2=Lior

|year=1971

|title=On the complemented subspaces problem

|journal=Israel Journal of Mathematics

|volume=9

|issue=2

|pages=263–269

|doi=10.1007/BF02771592 | doi-access=free}} The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer $n,$ any finite-dimensional normed space, with dimension sufficiently large compared to $n,$ contains subspaces nearly isometric to the $n$-dimensional Euclidean space.

The next result gives the solution of the so-called {{em|homogeneous space problem}}. An infinite-dimensional Banach space $X$ is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to $\backslash ell^2$ is homogeneous, and Banach asked for the converse.see p. 245 in {{harvtxt|Banach|1932}}. The homogeneity property is called "propriété (15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec $(L^2).$ possède la propriété (15)".

{{math theorem| name = TheoremGowers, W. T. (1996), "A new dichotomy for Banach spaces", Geom. Funct. Anal. 6:1083–1093. | math_statement = A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space.}}

An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces.

The Gowers dichotomy theorem asserts that every infinite-dimensional Banach space $X$ contains, either a subspace $Y$ with unconditional basis, or a hereditarily indecomposable subspace $Z,$ and in particular, $Z$ is not isomorphic to its closed hyperplanes.see {{cite journal|last1=Gowers|first1=W. T.|year=1994|title=A solution to Banach's hyperplane problem|journal=Bull. London Math. Soc.|volume=26|issue=6|pages=523–530|doi=10.1112/blms/26.6.523}}

If $X$ is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis,see {{cite journal|last1=Komorowski|first1=Ryszard A.|last2=Tomczak-Jaegermann|first2=Nicole|year=1995|title=Banach spaces without local unconditional structure|journal=Israel Journal of Mathematics|volume=89|issue=1–3|pages=205–226|arxiv=math/9306211|doi=10.1007/bf02808201|doi-access=free|s2cid=5220304}} and also {{cite journal|last1=Komorowski|first1=Ryszard A.|last2=Tomczak-Jaegermann|first2=Nicole|year=1998|title=Erratum to: Banach spaces without local unconditional structure|journal=Israel Journal of Mathematics|volume=105|pages=85–92|arxiv=math/9607205|doi=10.1007/bf02780323|doi-access=free|s2cid=18565676}} that $X$ is isomorphic to $\backslash ell^2.$

If $T\; :\; X\; \backslash to\; Y$ is an isometry from the Banach space $X$ onto the Banach space $Y$ (where both $X$ and $Y$ are vector spaces over $\backslash R$), then the Mazur–Ulam theorem states that $T$ must be an affine transformation.

In particular, if $T(0\_X)\; =\; 0\_Y,$ this is $T$ maps the zero of $X$ to the zero of $Y,$ then $T$ must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.

Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.

Anderson–Kadec theorem (1965–66) proves{{cite book|author=C. Bessaga, A. Pełczyński|title=Selected Topics in Infinite-Dimensional Topology|url=https://books.google.com/books?id=7n9sAAAAMAAJ|year=1975|publisher=Panstwowe wyd. naukowe|pages=177–230}} that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved{{cite book |author=H. Torunczyk |title=Characterizing Hilbert Space Topology |publisher=Fundamenta Mathematicae |year=1981 |pages=247–262}} that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

When two compact Hausdorff spaces $K\_1$ and $K\_2$ are homeomorphic, the Banach spaces $C(K\_1)$ and $C(K\_2)$ are isometric. Conversely, when $K\_1$ is not homeomorphic to $K\_2,$ the (multiplicative) Banach–Mazur distance between $C(K\_1)$ and $C(K\_2)$ must be greater than or equal to $2,$ see above the results by Amir and Cambern.

Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 2:150–156.

{{math theorem| name = TheoremMilutin. See also Rosenthal, Haskell P., "The Banach spaces C(K)" in Handbook of the geometry of Banach spaces, Vol. 2, 1547–1602, North-Holland, Amsterdam, 2003. | math_statement =Let $K$ be an uncountable compact metric space. Then $C(K)$ is isomorphic to $C([0,\; 1]).$}}

The situation is different for countably infinite compact Hausdorff spaces.

Every countably infinite compact $K$ is homeomorphic to some closed interval of ordinal numbers

$$\backslash langle\; 1,\; \backslash alpha\; \backslash rangle\; =\; \backslash \{\; \backslash gamma\; \backslash mid\; 1\; \backslash leq\; \backslash gamma\; \backslash leq\; \backslash alpha\backslash \}$$

equipped with the order topology, where $\backslash alpha$ is a countably infinite ordinal.One can take {{math|1=α = ω{{i sup|βn}}}}, where $\backslash beta\; +\; 1$ is the Cantor–Bendixson rank of $K,$ and $n\; >\; 0$ is the finite number of points in the $\backslash beta$-th derived set $K(\backslash beta)$ of $K.$ See Mazurkiewicz, Stefan; Sierpiński, Wacław (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Mathematicae 1: 17–27.

The Banach space $C(K)$ is then isometric to {{math|C(⟨1, α⟩)}}. When $\backslash alpha,\; \backslash beta$ are two countably infinite ordinals, and assuming $\backslash alpha\; \backslash leq\; \backslash beta,$ the spaces {{math|C(⟨1, α⟩)}} and {{math|C(⟨1, β⟩)}} are isomorphic if and only if {{math|β < α^ω}}.Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions", Studia Math. 19:53–62.

For example, the Banach spaces

$$C(\backslash langle\; 1,\; \backslash omega\backslash rangle),\; \backslash \; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega\}\; \backslash rangle),\; \backslash \; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega^2\}\backslash rangle),\; \backslash \; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega^3\}\; \backslash rangle),\; \backslash cdots,\; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega^\backslash omega\}\; \backslash rangle),\; \backslash cdots$$

are mutually non-isomorphic.

{{main|List of Banach spaces}}

{{ListOfBanachSpaces}}

{{clear}}

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details.

The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces.

Fréchet differentiability is a stronger condition than Gateaux differentiability.

The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions $\backslash R\; \backslash to\; \backslash R,$ or the space of all distributions on $\backslash R,$ are complete but are not normed vector spaces and hence not Banach spaces.

In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.

• {{annotated link|Space (mathematics)}}
• {{annotated link|Fréchet space}}
• {{annotated link|Hardy space}}
• {{annotated link|Hilbert space}}
• {{annotated link|L-semi-inner product}}
• {{annotated link|Lp space|$L^p$ space}}
• {{annotated link|Sobolev space}}
• {{annotated link|Banach lattice}}
• {{annotated link|Banach disk}}
• {{annotated link|Banach manifold}}
• {{annotated link|Banach bundle}}
• {{annotated link|Distortion problem}}
• {{annotated link|Interpolation space}}
• {{annotated link|Locally convex topological vector space}}
• {{annotated link|Modulus and characteristic of convexity}}
• {{annotated link|Smith space}}
• {{annotated link|Topological vector space}}
• {{annotated link|Tsirelson space}}

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{{Commons category|Banach spaces}}

• {{springer|title=Banach space|id=p/b015190}}
• {{MathWorld|BanachSpace|Banach Space}}

{{Banach spaces}}

{{Functional Analysis}}

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