Locally convex topological vector space

Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced).

After the notion of a general topological space was defined by Felix Hausdorff in 1914,Hausdorff, F. Grundzüge der Mengenlehre (1914) although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces.von Neumann, J. Collected works. Vol II. pp. 94–104Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).von Neumann, J. Collected works. Vol II. pp. 508–527Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2.

A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spacesBanach, S. Theory of linear operations p. 75. Ch. VIII. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932) (in which case the unit ball of the dual is metrizable).

Suppose $X$ is a vector space over $\backslash mathbb\{K\},$ a subfield of the complex numbers (normally $\backslash Complex$ itself or $\backslash R$).

A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

A topological vector space (TVS) is called {{em|{{visible anchor|locally convex}}}} if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets.{{sfn|Narici|Beckenstein|2011|pp=67–113}} The term {{em|{{visible anchor|locally convex topological vector space}}}} is sometimes shortened to {{em|{{visible anchor|locally convex space}}}} or {{abbreviation|{{em|{{visible anchor|LCTVS}}}}|Locally Convex Topological Vector Space}}.

A subset $C$ in $X$ is called

1. Convex if for all $x,\; y\; \backslash in\; C,$ and $0\; \backslash leq\; t\; \backslash leq\; 1,$ $tx\; +\; (1-t)y\; \backslash in\; C.$ In other words, $C$ contains all line segments between points in $C.$
2. Circled if for all $x\; \backslash in\; C$ and scalars $s,$ if $|s|\; =\; 1$ then $s\; x\; \backslash in\; C.$ If $\backslash mathbb\{K\}\; =\; \backslash R,$ this means that $C$ is equal to its reflection through the origin. For $\backslash mathbb\{K\}\; =\; \backslash Complex,$ it means for any $x\; \backslash in\; C,$ $C$ contains the circle through $x,$ centred on the origin, in the one-dimensional complex subspace generated by $x.$
3. Balanced if for all $x\; \backslash in\; C$ and scalars $s,$ if $|s|\; \backslash leq\; 1$ then $s\; x\; \backslash in\; C.$ If $\backslash mathbb\{K\}\; =\; \backslash R,$ this means that if $x\; \backslash in\; C,$ then $C$ contains the line segment between $x$ and $-x.$ For $\backslash mathbb\{K\}\; =\; \backslash Complex,$ it means for any $x\; \backslash in\; C,$ $C$ contains the disk with $x$ on its boundary, centred on the origin, in the one-dimensional complex subspace generated by $x.$ Equivalently, a balanced set is a "circled cone"{{Citation needed|date=August 2024}}. Note that in the TVS $\backslash mathbb\; R^2$, $x=(1,1)$ belongs to $C=(\{\}$ball centered at the origin of radius $\backslash sqrt2$$\{\})$, but $2x=(2,2)$ does not belong; indeed, C is {{em|not}} a cone, but {{em|is}} balanced.
4. A cone (when the underlying field is ordered) if for all $x\; \backslash in\; C$ and $t\; \backslash geq\; 0\; ,$ $t\; x\; \backslash in\; C.$
5. Absorbent or absorbing if for every $x\; \backslash in\; X,$ there exists $r\; >\; 0$ such that $x\; \backslash in\; t\; C$ for all $t\; \backslash in\; \backslash mathbb\{K\}$ satisfying $|t|\; >\; r.$ The set $C$ can be scaled out by any "large" value to absorb every point in the space.
6. * In any TVS, every neighborhood of the origin is absorbent.{{sfn|Narici|Beckenstein|2011|pp=67–113}}
7. Absolutely convex or a {{em|{{visible anchor|disk}}}} if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to $\backslash leq\; 1$; such a set is absorbent if it spans all of $X.$

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of {{em|{{visible anchor|absolutely convex}}}} sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets.{{sfn|Narici|Beckenstein|2011|pp=|p=83}}

Every TVS has a neighborhood basis at the origin consisting of balanced sets, but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced {{em|and}} convex. It is possible for a TVS to have {{em|some}} neighborhoods of the origin that are convex and yet not be locally convex because it has no neighborhood basis at the origin consisting entirely of convex sets (that is, every neighborhood basis at the origin contains some non-convex set); for example, every non-locally convex TVS $X$ has itself (that is, $X$) as a convex neighborhood of the origin.

Because translation is continuous (by definition of topological vector space), all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

A seminorm on $X$ is a map $p\; :\; X\; \backslash to\; \backslash R$ such that

1. $p$ is nonnegative or positive semidefinite: $p(x)\; \backslash geq\; 0$;
2. $p$ is positive homogeneous or positive scalable: $p(s\; x)\; =\; |s|\; p(x)$ for every scalar $s.$ So, in particular, $p(0)\; =\; 0$;
3. $p$ is subadditive. It satisfies the triangle inequality: $p(x\; +\; y)\; \backslash leq\; p(x)\; +\; p(y).$

If $p$ satisfies positive definiteness, which states that if $p(x)\; =\; 0$ then $x\; =\; 0,$ then $p$ is a norm.

While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

If $X$ is a vector space and $\backslash mathcal\{P\}$ is a family of seminorms on $X$ then a subset $\backslash mathcal\{Q\}$ of $\backslash mathcal\{P\}$ is called a base of seminorms for $\backslash mathcal\{P\}$ if for all $p\; \backslash in\; \backslash mathcal\{P\}$ there exists a $q\; \backslash in\; \backslash mathcal\{Q\}$ and a real $r\; >\; 0$ such that $p\; \backslash leq\; r\; q.${{sfn|Narici|Beckenstein|2011|p=122}}

Definition (second version): A locally convex space is defined to be a vector space $X$ along with a family $\backslash mathcal\{P\}$ of seminorms on $X.$

Suppose that $X$ is a vector space over $\backslash mathbb\{K\},$ where $\backslash mathbb\{K\}$ is either the real or complex numbers.

A family of seminorms $\backslash mathcal\{P\}$ on the vector space $X$ induces a canonical vector space topology on $X$, called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on $X$ for which all maps in $\backslash mathcal\{P\}$ are continuous.

It is possible for a locally convex topology on a space $X$ to be induced by a family of norms but for $X$ to {{em|not}} be normable (that is, to have its topology be induced by a single norm).

An open set in $\backslash mathbb\{R\}\_\{\backslash geq0\}$ has the form $[0,\; r)$, where $r$ is a positive real number. The family of preimages as $p$ ranges over a family of seminorms $\backslash mathcal\{P\}$ and $r$ ranges over the positive real numbers

is a subbasis at the origin for the topology induced by $\backslash mathcal\{P\}$. These sets are convex, as follows from properties 2 and 3 of seminorms.

Intersections of finitely many such sets are then also convex, and since the collection of all such finite intersections is a basis at the origin it follows that the topology is locally convex in the sense of the {{em|first}} definition given above.

Recall that the topology of a TVS is translation invariant, meaning that if $S$ is any subset of $X$ containing the origin then for any $x\; \backslash in\; X,$ $S$ is a neighborhood of the origin if and only if $x\; +\; S$ is a neighborhood of $x$;

thus it suffices to define the topology at the origin.

A base of neighborhoods of $y$ for this topology is obtained in the following way: for every finite subset $F$ of $\backslash mathcal\{P\}$ and every $r\; >\; 0,$ let

If $X$ is a locally convex space and if $\backslash mathcal\{P\}$ is a collection of continuous seminorms on $X$, then $\backslash mathcal\{P\}$ is called a base of continuous seminorms if it is a base of seminorms for the collection of {{em|all}} continuous seminorms on $X$.{{sfn|Narici|Beckenstein|2011|p=122}} Explicitly, this means that for all continuous seminorms $p$ on $X$, there exists a $q\; \backslash in\; \backslash mathcal\{P\}$ and a real $r\; >\; 0$ such that $p\; \backslash leq\; r\; q.${{sfn|Narici|Beckenstein|2011|p=122}}

If $\backslash mathcal\{P\}$ is a base of continuous seminorms for a locally convex TVS $X$ then the family of all sets of the form as $q$ varies over $\backslash mathcal\{P\}$ and $r$ varies over the positive real numbers, is a {{em|base}} of neighborhoods of the origin in $X$ (not just a subbasis, so there is no need to take finite intersections of such sets).{{sfn|Narici|Beckenstein|2011|p=122}}Let be the open unit ball associated with the seminorm $p$ and note that if $r\; >\; 0$ is real then and so $\backslash tfrac\{1\}\{r\}\; V\_p\; =\; V\_\{r\; p\}.$ Thus a basic open neighborhood of the origin induced by $\backslash mathcal\{P\}$ is a finite intersection of the form $V\_\{r\_1\; p\_1\}\; \backslash cap\; \backslash cdots\; \backslash cap\; V\_\{r\_n\; p\_n\}$ where $p\_1,\; \backslash ldots,\; p\_n\; \backslash in\; \backslash mathcal\{P\}$ and $r\_1,\; \backslash ldots,\; r\_n$ are all positive reals. Let $p\; :=\; \backslash max\; \backslash left\backslash \{r\_1\; p\_1,\; \backslash ldots,\; r\_n\; p\_n\backslash right\backslash \},$ which is a continuous seminorm and moreover, $V\_p\; =\; V\_\{r\_1\; p\_1\}\; \backslash cap\; \backslash cdots\; \backslash cap\; V\_\{r\_n\; p\_n\}.$ Pick $r\; >\; 0$ and $q\; \backslash in\; \backslash mathcal\{P\}$ such that $p\; \backslash leq\; r\; q,$ where this inequality holds if and only if $V\_\{r\; q\}\; \backslash subseteq\; V\_p.$ Thus $\backslash tfrac\{1\}\{r\}\; V\_q\; =\; V\_\{r\; q\}\; \backslash subseteq\; V\_p\; =\; V\_\{r\_1\; p\_1\}\; \backslash cap\; \backslash cdots\; \backslash cap\; V\_\{r\_n\; p\_n\},$ as desired.

A family $\backslash mathcal\{P\}$ of seminorms on a vector space $X$ is called saturated if for any $p$ and $q$ in $\backslash mathcal\{P\},$ the seminorm defined by $x\; \backslash mapsto\; \backslash max\; \backslash \{\; p(x),\; q(x)\; \backslash \}$ belongs to $\backslash mathcal\{P\}.$

If $\backslash mathcal\{P\}$ is a saturated family of continuous seminorms that induces the topology on $X$ then the collection of all sets of the form as $p$ ranges over $\backslash mathcal\{P\}$ and $r$ ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets;{{sfn|Narici|Beckenstein|2011|p=122}}

This forms a basis at the origin rather than merely a subbasis so that in particular, there is {{em|no}} need to take finite intersections of such sets.{{sfn|Narici|Beckenstein|2011|p=122}}

The following theorem implies that if $X$ is a locally convex space then the topology of $X$ can be a defined by a family of continuous {{em|norms}} on $X$ (a norm is a seminorm $s$ where $s(x)=0$ implies $x=0$) if and only if there exists {{em|at least one}} continuous {{em|norm}} on $X$.{{sfn|Jarchow|1981|p=130}} This is because the sum of a norm and a seminorm is a norm so if a locally convex space is defined by some family $\backslash mathcal\{P\}$ of seminorms (each of which is necessarily continuous) then the family $\backslash mathcal\{P\}\; +\; n\; :=\; \backslash \{p\; +\; n\; :\; p\; \backslash in\; \backslash mathcal\{P\}\backslash \}$ of (also continuous) norms obtained by adding some given continuous norm $n$ to each element, will necessarily be a family of norms that defines this same locally convex topology.

If there exists a continuous norm on a topological vector space $X$ then $X$ is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

{{Math theorem|name=Theorem{{sfn|Jarchow|1981|pp=129–130}}|math_statement=

Let $X$ be a Fréchet space over the field $\backslash mathbb\{K\}.$

Then the following are equivalent:

1. $X$ does {{em|not}} admit a continuous norm (that is, any continuous seminorm on $X$ can {{em|not}} be a norm).
2. $X$ contains a vector subspace that is TVS-isomorphic to $\backslash mathbb\{K\}^\{\backslash N\}.$
3. $X$ contains a complemented vector subspace that is TVS-isomorphic to $\backslash mathbb\{K\}^\{\backslash N\}.$

}}

Suppose that the topology of a locally convex space $X$ is induced by a family $\backslash mathcal\{P\}$ of continuous seminorms on $X$.

If $x\; \backslash in\; X$ and if $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a net in $X$, then $x\_\{\backslash bull\}\; \backslash to\; x$ in $X$ if and only if for all $p\; \backslash in\; \backslash mathcal\{P\},$ $p\backslash left(x\_\{\backslash bull\}\; -\; x\backslash right)\; =\; \backslash left(p\backslash left(x\_i\backslash right)\; -\; x\backslash right)\_\{i\; \backslash in\; I\}\; \backslash to\; 0.${{sfn|Narici|Beckenstein|2011|p=126}}

Moreover, if $x\_\{\backslash bull\}$ is Cauchy in $X$, then so is $p\backslash left(x\_\{\backslash bull\}\backslash right)\; =\; \backslash left(p\backslash left(x\_i\backslash right)\backslash right)\_\{i\; \backslash in\; I\}$ for every $p\; \backslash in\; \backslash mathcal\{P\}.${{sfn|Narici|Beckenstein|2011|p=126}}

Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice.

The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge.

The key feature of seminorms which ensures the convexity of their $\backslash varepsilon$-balls is the triangle inequality.

For an absorbing set $C$ such that if $x\; \backslash in\; C,$ then $t\; x\; \backslash in\; C$ whenever $0\; \backslash leq\; t\; \backslash leq\; 1,$ define the Minkowski functional of $C$ to be

$$\backslash mu\_C(x)\; =\; \backslash inf\; \backslash \{r\; >\; 0:\; x\; \backslash in\; r\; C\backslash \}.$$

From this definition it follows that $\backslash mu\_C$ is a seminorm if $C$ is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets

form a base of convex absorbent balanced sets.

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=67–113}}|math_statement=

Suppose that $X$ is a (real or complex) vector space and let $\backslash mathcal\{B\}$ be a filter base of subsets of $X$ such that:

1. Every $B\; \backslash in\; \backslash mathcal\{B\}$ is convex, balanced, and absorbing;
2. For every $B\; \backslash in\; \backslash mathcal\{B\}$ there exists some real $r$ satisfying such that $r\; B\; \backslash in\; \backslash mathcal\{B\}.$

Then $\backslash mathcal\{B\}$ is a neighborhood base at 0 for a locally convex TVS topology on $X.$

}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=67–113}}|math_statement=

Suppose that $X$ is a (real or complex) vector space and let $\backslash mathcal\{L\}$ be a non-empty collection of convex, balanced, and absorbing subsets of $X.$

Then the set of all positive scalar multiples of finite intersections of sets in $\backslash mathcal\{L\}$ forms a neighborhood base at the origin for a locally convex TVS topology on $X.$

}}

Example: auxiliary normed spaces

If $W$ is convex and absorbing in $X$ then the symmetric set $D\; :=\; \backslash bigcap\_\{|u|=1\}\; u\; W$ will be convex and balanced (also known as an {{em|absolutely convex set}} or a {{em|disk}}) in addition to being absorbing in $X.$

This guarantees that the Minkowski functional $p\_D\; :\; X\; \backslash to\; \backslash R$ of $D$ will be a seminorm on $X,$ thereby making $\backslash left(X,\; p\_D\backslash right)$ into a seminormed space that carries its canonical pseudometrizable topology. The set of scalar multiples $r\; D$ as $r$ ranges over $\backslash left\backslash \{\backslash tfrac\{1\}\{2\},\; \backslash tfrac\{1\}\{3\},\; \backslash tfrac\{1\}\{4\},\; \backslash ldots\backslash right\backslash \}$ (or over any other set of non-zero scalars having $0$ as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If $X$ is a topological vector space and if this convex absorbing subset $W$ is also a bounded subset of $X,$ then the absorbing disk $D\; :=\; \backslash bigcap\_\{|u|=1\}\; u\; W$ will also be bounded, in which case $p\_D$ will be a norm and $\backslash left(X,\; p\_D\backslash right)$ will form what is known as an auxiliary normed space. If this normed space is a Banach space then $D$ is called a {{em|Banach disk}}.

• A family of seminorms $\backslash left(p\_\{\backslash alpha\}\backslash right)\_\{\backslash alpha\}$ is called total or separated or is said to separate points if whenever $p\_\{\backslash alpha\}(x)\; =\; 0$ holds for every $\backslash alpha$ then $x$ is necessarily $0.$ A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
• A pseudometric is a generalization of a metric which does not satisfy the condition that $d(x,\; y)\; =\; 0$ only when $x\; =\; y.$ A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by $$d(x,y)=\backslash sum^\backslash infty\_n\; \backslash frac\{1\}\{2^n\}\; \backslash frac\{p\_n(x-y)\}\{1+p\_n(x-y)\}$$ (where the $1/2^n$ can be replaced by any positive summable sequence $a\_n$). This pseudometric is translation-invariant, but not homogeneous, meaning $d(k\; x,\; k\; y)\; \backslash neq\; |k|\; d(x,\; y),$ and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
• As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
• A Cauchy net in a locally convex space is a net $\backslash left(x\_a\backslash right)\_\{a\; \backslash in\; A\}$ such that for every $r\; >\; 0$ and every seminorm $p\_\backslash alpha,$ there exists some index $c\; \backslash in\; A$ such that for all indices $a,\; b\; \backslash geq\; c,$ In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.
• A family of seminorms becomes a preordered set under the relation $p\_\backslash alpha\; \backslash leq\; p\_\backslash beta$ if and only if there exists an $M\; >\; 0$ such that for all $x,$ $p\_\backslash alpha(x)\; \backslash leq\; M\; p\_\backslash beta(x).$ One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every $\backslash alpha$ and $\backslash beta,$ there is a $\backslash gamma$ such that $p\_\backslash alpha\; +\; p\_\backslash beta\; \backslash leq\; p\_\backslash gamma.$ Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family $\backslash left(p\_\backslash alpha(x)\backslash right)\_\{\backslash alpha\; \backslash in\; I\},$ let $\backslash Phi$ be the set of finite subsets of $I$ and then for every $F\; \backslash in\; \backslash Phi$ define $$q\_F\; =\; \backslash sum\_\{\backslash alpha\; \backslash in\; F\}\; p\_\{\backslash alpha\}.$$ One may check that $\backslash left(q\_F\backslash right)\_\{F\; \backslash in\; \backslash Phi\}$ is an equivalent directed family.
• If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if the origin has a bounded neighborhood.

Let $X$ be a TVS.

Say that a vector subspace $M$ of $X$ has the extension property if any continuous linear functional on $M$ can be extended to a continuous linear functional on $X$.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

Say that $X$ has the Hahn-Banach extension property (HBEP) if every vector subspace of $X$ has the extension property.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.

For complete metrizable TVSs there is a converse:

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=225–273}}|note=Kalton|math_statement=

Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.

}}

If a vector space $X$ has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

{{See also|Topological vector space#Properties}}

Throughout, $\backslash mathcal\{P\}$ is a family of continuous seminorms that generate the topology of $X.$

Topological closure

If $S\; \backslash subseteq\; X$ and $x\; \backslash in\; X,$ then $x\; \backslash in\; \backslash operatorname\{cl\}\; S$ if and only if for every $r\; >\; 0$ and every finite collection $p\_1,\; \backslash ldots,\; p\_n\; \backslash in\; \backslash mathcal\{P\}$ there exists some $s\; \backslash in\; S$ such that {{sfn|Narici|Beckenstein|2011|p=149}}

The closure of $\backslash \{0\backslash \}$ in $X$ is equal to $\backslash bigcap\_\{p\; \backslash in\; \backslash mathcal\{P\}\}\; p^\{-1\}(0).${{sfn|Narici|Beckenstein|2011|pp=149–153}}

Topology of Hausdorff locally convex spaces

Every Hausdorff locally convex space is homeomorphic to a vector subspace of a product of Banach spaces.{{sfn|Narici|Beckenstein|2011|pp=115–154}}

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

Algebraic properties of convex subsets

A subset $C$ is convex if and only if $t\; C\; +\; (1\; -\; t)\; C\; \backslash subseteq\; C$ for all $0\; \backslash leq\; t\; \backslash leq\; 1${{sfn|Rudin|1991|p=6}} or equivalently, if and only if $(s\; +\; t)\; C\; =\; s\; C\; +\; t\; C$ for all positive real $s\; >\; 0\; \backslash text\{\; and\; \}\; t\; >\; 0,${{sfn|Rudin|1991|p=38}} where because $(s\; +\; t)\; C\; \backslash subseteq\; s\; C\; +\; t\; C$ always holds, the equals sign $\backslash ,=\backslash ,$ can be replaced with $\backslash ,\backslash supseteq.\backslash ,$ If $C$ is a convex set that contains the origin then $C$ is star shaped at the origin and for all non-negative real $s\; \backslash geq\; 0\; \backslash text\{\; and\; \}\; t\; \backslash geq\; 0,$ $(s\; C)\; \backslash cap\; (t\; C)\; =\; (\backslash min\_\{\}\; \backslash \{s,\; t\backslash \})\; C.$

The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.{{sfn|Trèves|2006|p=126}}

Topological properties of convex subsets

• Suppose that $Y$ is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of $Y$ are exactly those that are of the form for some $z\; \backslash in\; Y$ and some positive continuous sublinear functional $p$ on $Y.${{sfn|Narici|Beckenstein|2011|pp=177–220}}
• The interior and closure of a convex subset of a TVS is again convex.{{sfn|Trèves|2006|p=126}}
• If $C$ is a convex set with non-empty interior, then the closure of $C$ is equal to the closure of the interior of $C$; furthermore, the interior of $C$ is equal to the interior of the closure of $C.${{sfn|Trèves|2006|p=126}}{{sfn|Schaefer|Wolff|1999|p=38}}
• So if the interior of a convex set $C$ is non-empty then $C$ is a closed (respectively, open) set if and only if it is a regular closed (respectively, regular open) set.
• If $C$ is convex and then{{sfn|Jarchow|1981|pp=101-104}} $t\; \backslash operatorname\{Int\}\; C\; +\; (1\; -\; t)\; \backslash operatorname\{cl\}\; C\; ~\backslash subseteq~\; \backslash operatorname\{Int\}\; C.$ Explicitly, this means that if $C$ is a convex subset of a TVS $X$ (not necessarily Hausdorff or locally convex), $y$ belongs to the closure of $C,$ and $x$ belongs to the interior of $C,$ then the open line segment joining $x$ and $y$ belongs to the interior of $C;$ that is, {{sfn|Schaefer|Wolff|1999|p=38}}{{sfn|Conway|1990|p=102}}Fix so it remains to show that $w\_0\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; r\; x\; +\; (1\; -\; r)\; y$ belongs to $\backslash operatorname\{int\}\_X\; C.$ By replacing $C,\; x,\; y$ with $C\; -\; w\_0,\; x\; -\; w\_0,\; y\; -\; w\_0$ if necessary, we may assume without loss of generality that $r\; x\; +\; (1\; -\; r)\; y\; =\; 0,$ and so it remains to show that $C$ is a neighborhood of the origin. Let so that $y\; =\; \backslash tfrac\{r\}\{r\; -\; 1\}\; x\; =\; s\; x.$ Since scalar multiplication by $s\; \backslash neq\; 0$ is a linear homeomorphism $X\; \backslash to\; X,$ $\backslash operatorname\{cl\}\_X\; \backslash left(\backslash tfrac\{1\}\{s\}\; C\backslash right)\; =\; \backslash tfrac\{1\}\{s\}\; \backslash operatorname\{cl\}\_X\; C.$ Since $x\; \backslash in\; \backslash operatorname\{int\}\; C$ and $y\; \backslash in\; \backslash operatorname\{cl\}\; C,$ it follows that $x\; =\; \backslash tfrac\{1\}\{s\}\; y\; \backslash in\; \backslash operatorname\{cl\}\; \backslash left(\backslash tfrac\{1\}\{s\}\; C\backslash right)\; \backslash cap\; \backslash operatorname\{int\}\; C$ where because $\backslash operatorname\{int\}\; C$ is open, there exists some $c\_0\; \backslash in\; \backslash left(\backslash tfrac\{1\}\{s\}\; C\backslash right)\; \backslash cap\; \backslash operatorname\{int\}\; C,$ which satisfies $s\; c\_0\; \backslash in\; C.$ Define $h\; :\; X\; \backslash to\; X$ by $x\; \backslash mapsto\; r\; x\; +\; (1\; -\; r)\; s\; c\_0\; =\; r\; x\; -\; r\; c\_0,$ which is a homeomorphism because The set $h\backslash left(\backslash operatorname\{int\}\; C\backslash right)$ is thus an open subset of $X$ that moreover contains $h(c\_0)\; =\; r\; c\_0\; -\; r\; c\_0\; =\; 0.$ If $c\; \backslash in\; \backslash operatorname\{int\}\; C$ then $h(c)\; =\; r\; c\; +\; (1\; -\; r)\; s\; c\_0\; \backslash in\; C$ since $C$ is convex, and $s\; c\_0,\; c\; \backslash in\; C,$ which proves that $h\backslash left(\backslash operatorname\{int\}\; C\backslash right)\; \backslash subseteq\; C.$ Thus $h\backslash left(\backslash operatorname\{int\}\; C\backslash right)$ is an open subset of $X$ that contains the origin and is contained in $C.$ Q.E.D.
• If $M$ is a closed vector subspace of a (not necessarily Hausdorff) locally convex space$X,$ $V$ is a convex neighborhood of the origin in $M,$ and if $z\; \backslash in\; X$ is a vector {{em|not}} in $V,$ then there exists a convex neighborhood $U$ of the origin in $X$ such that $V\; =\; U\; \backslash cap\; M$ and $z\; \backslash not\backslash in\; U.${{sfn|Trèves|2006|p=126}}
• The closure of a convex subset of a locally convex Hausdorff space $X$ is the same for {{em|all}} locally convex Hausdorff TVS topologies on $X$ that are compatible with duality between $X$ and its continuous dual space.{{sfn|Trèves|2006|p=370}}
• In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.{{sfn|Narici|Beckenstein|2011|pp=67–113}}
• In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.{{sfn|Narici|Beckenstein|2011|pp=67–113}}
• More generally, if $K$ is a compact subset of a locally convex space, then the convex hull $\backslash operatorname\{co\}\; K$ (respectively, the disked hull $\backslash operatorname\{cobal\}\; K$) is compact if and only if it is complete.{{sfn|Narici|Beckenstein|2011|pp=67–113}}
• In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.{{sfn|Narici|Beckenstein|2011|pp=155–176}}
• In a Fréchet space, the closed convex hull of a compact set is compact.{{sfn|Rudin|1991|p=7}}
• In a locally convex space, any linear combination of totally bounded sets is totally bounded.{{sfn|Narici|Beckenstein|2011|pp=155–176}}

For any subset $S$ of a TVS $X,$ the convex hull (respectively, closed convex hull, balanced hull, convex balanced hull) of $S,$ denoted by $\backslash operatorname\{co\}\; S$ (respectively, $\backslash overline\{\backslash operatorname\{co\}\}\; S,$ $\backslash operatorname\{bal\}\; S,$ $\backslash operatorname\{cobal\}\; S$), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of $X$ containing $S.$

• The convex hull of compact subset of a Hilbert space is {{em|not}} necessarily closed and so also {{em|not}} necessarily compact. For example, 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 $e\_n\; =\; (0,\; \backslash ldots,\; 0,\; 1,\; 0,\; \backslash ldots)$ be the standard orthonormal basis (that is $1$ at the $n^\{\backslash text\{th\}\}$-coordinate). The closed set $S\; =\; \backslash \{0\backslash \}\; \backslash cup\; \backslash left\backslash \{\backslash tfrac\{1\}\{1\}\; e\_n,\; \backslash tfrac\{1\}\{2\}\; e\_2,\; \backslash tfrac\{1\}\{3\}\; e\_3,\; \backslash ldots\backslash right\backslash \}$ is compact but its convex hull $\backslash operatorname\{co\}\; S$ is {{em|not}} a closed set because $h\; :=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash tfrac\{1\}\{2^n\}\; \backslash tfrac\{1\}\{n\}\; e\_n$ belongs to the closure of $\backslash operatorname\{co\}\; S$ in $H$ but $h\; \backslash not\backslash in\backslash operatorname\{co\}\; S$ (since every sequence $z\; \backslash in\; \backslash operatorname\{co\}\; S$ is a finite convex combination of elements of $S$ and so is necessarily $0$ in all but finitely many coordinates, which is not true of $h$).{{sfn|Aliprantis|Border|2006|p=185}} 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 $X\; :=\; \backslash operatorname\{span\}\; S$ 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 compact subset might {{em|fail}} to be compact (although it will be precompact/totally bounded).
• In a Hausdorff locally convex space $X,$ the closed convex hull $\backslash overline\{\backslash operatorname\{co\}\}^X\; S\; =\; \backslash operatorname\{cl\}\_X\; \backslash operatorname\{co\}\; S$ of compact subset $S$ is not necessarily compact although it is a precompact (also called "totally bounded") subset, which means that its closure, {{em|when taken in a completion}} $\backslash widehat\{X\}$ of $X,$ will be compact (here $X\; \backslash subseteq\; \backslash widehat\{X\},$ so that $X\; =\; \backslash widehat\{X\}$ if and only if $X$ is complete); that is to say, $\backslash operatorname\{cl\}\_\{\backslash widehat\{X\}\}\; \backslash overline\{\backslash operatorname\{co\}\}^X\; S$ will be compact. So for example, the closed convex hull $C\; :=\; \backslash overline\{\backslash operatorname\{co\}\}^X\; S$ of a compact subset of $S$ of a pre-Hilbert space $X$ is always a precompact subset of $X,$ and so the closure of $C$ in any Hilbert space $H$ containing $X$ (such as the Hausdorff completion of $X$ for instance) will be compact (this is the case in the previous example above).
• In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is again compact.
• In a Hausdorff locally convex TVS, the convex hull of a precompact set is again precompact.{{sfn|Trèves|2006|p=67}} Consequently, in a complete Hausdorff locally convex space, the closed convex hull of a compact subset is again compact.{{sfn|Trèves|2006|p=145}}
• In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).{{sfn|Narici|Beckenstein|2011|pp=67–113}}
• This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is {{em|closed}} (in addition to being compact{{sfn|Rudin|1991|pp=72-73}} and convex); in particular, the convex hull of such a union is equal to the {{em|closed}} convex hull of that union.
• In general, the closed convex hull of a compact set is not necessarily compact. However, every compact subset of $\backslash Reals^n$ (where ) does have a compact convex hull.{{sfn|Rudin|1991|pp=72-73}}
• In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but {{em|not}} closed.
• The bipolar theorem states that the bipolar (that is, the polar of the polar) of a subset of a locally convex Hausdorff TVS is equal to the closed convex balanced hull of that set.{{sfn|Trèves|2006|p=362}}
• The balanced hull of a convex set is {{em|not}} necessarily convex.
• If $C$ and $D$ are convex subsets of a topological vector space $X$ and if $x\backslash in\; \backslash operatorname\{co\}\; (C\; \backslash cup\; D),$ then there exist $c\; \backslash in\; C,$ $d\; \backslash in\; D,$ and a real number $r$ satisfying $0\; \backslash leq\; r\; \backslash leq\; 1$ such that $x\; =\; r\; c\; +\; (1\; -\; r)\; d.${{sfn|Trèves|2006|p=126}}
• If $M$ is a vector subspace of a TVS $X,$ $C$ a convex subset of $M,$ and $D$ a convex subset of $X$ such that $D\; \backslash cap\; M\; \backslash subseteq\; C,$ then $C\; =\; M\; \backslash cap\; \backslash operatorname\{co\}\; (C\; \backslash cup\; D).${{sfn|Trèves|2006|p=126}}
• Recall that the smallest balanced subset of $X$ containing a set $S$ is called the balanced hull of $S$ and is denoted by $\backslash operatorname\{bal\}\; S.$ For any subset $S$ of $X,$ the convex balanced hull of $S,$ denoted by $\backslash operatorname\{cobal\}\; S,$ is the smallest subset of $X$ containing $S$ that is convex and balanced.{{sfn|Trèves|2006|p=68}} The convex balanced hull of $S$ is equal to the convex hull of the balanced hull of $S$ (i.e. $\backslash operatorname\{cobal\}\; S\; =\; \backslash operatorname\{co\}\; (\backslash operatorname\{bal\}\; S)$), but the convex balanced hull of $S$ is {{em|not}} necessarily equal to the balanced hull of the convex hull of $S$ (that is, $\backslash operatorname\{cobal\}\; S$ is not necessarily equal to $\backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)$).{{sfn|Trèves|2006|p=68}}
• If $A,\; B\; \backslash subseteq\; X$ are subsets of a TVS $X$ and if $s$ is a scalar then $\backslash operatorname\{co\}\; (A\; +\; B)\; =\; \backslash operatorname\{co\}\; (A)\; +\; \backslash operatorname\{co\}\; (B),$ $\backslash operatorname\{co\}\; (s\; A)\; =\; s\; \backslash operatorname\{co\}\; A,${{sfn|Narici|Beckenstein|2011|p=108}} $\backslash operatorname\{co\}\; (A\; \backslash cup\; B)\; =\; \backslash operatorname\{co\}\; (A)\; \backslash cup\; \backslash operatorname\{co\}\; (B),$ and $\backslash overline\{\backslash operatorname\{co\}\}(s\; A)\; =\; s\; \backslash overline\{\backslash operatorname\{co\}\}(A).$ Moreover, if $\backslash overline\{\backslash operatorname\{co\}\}(A)$ is compact then $\backslash overline\{\backslash operatorname\{co\}\}(A\; +\; B)\; =\; \backslash overline\{\backslash operatorname\{co\}\}(A)\; +\; \backslash overline\{\backslash operatorname\{co\}\}(B).${{sfn|Dunford|1988|p=415}} However, the convex hull of a closed set need not be closed;{{sfn|Narici|Beckenstein|2011|p=108}} for example, the set is closed in $X\; :=\; \backslash R^2$ but its convex hull is the open set $\backslash left(-\backslash tfrac\{\backslash pi\}\{2\},\; \backslash tfrac\{\backslash pi\}\{2\}\backslash right)\; \backslash times\; \backslash R.$
• If $A,\; B\; \backslash subseteq\; X$ are subsets of a TVS $X$ whose closed convex hulls are compact, then $\backslash overline\{\backslash operatorname\{co\}\}(A\; \backslash cup\; B)\; =\; \backslash overline\{\backslash operatorname\{co\}\}\backslash left(\backslash overline\{\backslash operatorname\{co\}\}(A)\; \backslash cup\; \backslash overline\{\backslash operatorname\{co\}\}(B)\backslash right).${{sfn|Dunford|1988|p=415}}
• If $S$ is a convex set in a complex vector space $X$ and there exists some $z\; \backslash in\; X$ such that $z,\; iz,\; -z,\; -iz\; \backslash in\; S,$ then $r\; z\; +\; s\; i\; z\; \backslash in\; S$ for all real $r,\; s$ such that $|r|\; +\; |s|\; \backslash leq\; 1.$ In particular, $a\; z\; \backslash in\; S$ for all scalars $a$ such that $|a|^2\; \backslash leq\; \backslash tfrac\{1\}\{2\}.$
• Carathéodory's theorem: If $S$ is {{em|any}} subset of $\backslash Reals^n$ (where ) then for every $x\; \backslash in\; \backslash operatorname\{co\}\; S,$ there exist a finite subset $F\; \backslash subseteq\; S$ containing at most $n\; +\; 1$ points whose convex hull contains $x$ (that is, $|F|\; \backslash leq\; n\; +\; 1$ and $x\; \backslash in\; \backslash operatorname\{co\}\; F$).{{sfn|Rudin|1991|pp=73-74}}

Any vector space $X$ endowed with the trivial topology (also called the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology).

This topology is Hausdorff if and only $X\; =\; \backslash \{0\backslash \}.$

The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.

In contrast, the discrete topology forms a vector topology on $X$ if and only $X\; =\; \backslash \{0\backslash \}.$

This follows from the fact that every topological vector space is a connected space.

If $X$ is a real or complex vector space and if $\backslash mathcal\{P\}$ is the set of all seminorms on $X$ then the locally convex TVS topology, denoted by $\backslash tau\_\{\backslash operatorname\{lc\}\},$ that $\backslash mathcal\{P\}$ induces on $X$ is called the {{visible anchor|finest locally convex topology}} on $X.${{sfn|Narici|Beckenstein|2011|pp=125–126}}

This topology may also be described as the TVS-topology on $X$ having as a neighborhood base at the origin the set of all absorbing disks in $X.${{sfn|Narici|Beckenstein|2011|pp=125–126}}

Any locally convex TVS-topology on $X$ is necessarily a subset of $\backslash tau\_\{\backslash operatorname\{lc\}\}.$

$\backslash left(X,\; \backslash tau\_\{\backslash operatorname\{lc\}\}\backslash right)$ is Hausdorff.{{sfn|Narici|Beckenstein|2011|pp=149–153}}

Every linear map from $\backslash left(X,\; \backslash tau\_\{\backslash operatorname\{lc\}\}\backslash right)$ into another locally convex TVS is necessarily continuous.{{sfn|Narici|Beckenstein|2011|pp=149–153}}

In particular, every linear functional on $\backslash left(X,\; \backslash tau\_\{\backslash operatorname\{lc\}\}\backslash right)$ is continuous and every vector subspace of $X$ is closed in $\backslash left(X,\; \backslash tau\_\{\backslash operatorname\{lc\}\}\backslash right)$;{{sfn|Narici|Beckenstein|2011|pp=149–153}}

therefore, if $X$ is infinite dimensional then $\backslash left(X,\; \backslash tau\_\{\backslash operatorname\{lc\}\}\backslash right)$ is not pseudometrizable (and thus not metrizable).{{sfn|Narici|Beckenstein|2011|pp=125–126}}

Moreover, $\backslash tau\_\{\backslash operatorname\{lc\}\}$ is the {{em|only}} Hausdorff locally convex topology on $X$ with the property that any linear map from it into any Hausdorff locally convex space is continuous.{{sfn|Narici|Beckenstein|2011|p=476}}

The space $\backslash left(X,\; \backslash tau\_\{\backslash operatorname\{lc\}\}\backslash right)$ is a bornological space.{{sfn|Narici|Beckenstein|2011|p=446}}

Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalizes parts of the theory of normed spaces.

The family of seminorms can be taken to be the single norm.

Every Banach space is a complete Hausdorff locally convex space, in particular, the $L^p$ spaces with $p\; \backslash geq\; 1$ are locally convex.

More generally, every Fréchet space is locally convex.

A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.

The space $\backslash R^\{\backslash omega\}$ of real valued sequences with the family of seminorms given by

$$p\_i\; \backslash left(\backslash left\backslash \{x\_n\backslash right\backslash \}\_n\backslash right)\; =\; \backslash left|x\_i\backslash right|,\; \backslash qquad\; i\; \backslash in\; \backslash N$$

is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. This is also the limit topology of the spaces $\backslash R^n,$ embedded in $\backslash R^\{\backslash omega\}$ in the natural way, by completing finite sequences with infinitely many $0.$

Given any vector space $X$ and a collection $F$ of linear functionals on it, $X$ can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in $F$ continuous. This is known as the weak topology or the initial topology determined by $F.$

The collection $F$ may be the algebraic dual of $X$ or any other collection.

The family of seminorms in this case is given by $p\_f(x)\; =\; |f(x)|$ for all $f$ in $F.$

Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ such that where $a$ and $B$ are multiindices.

The family of seminorms defined by $p\_\{a,b\}(f)\; =\; \backslash sup\_x\; \backslash left|x^a\; D\_b\; f(x)\backslash right|$ is separated, and countable, and the space is complete, so this metrizable space is a Fréchet space.

It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions.

An important function space in functional analysis is the space $D(U)$ of smooth functions with compact support in $U\; \backslash subseteq\; \backslash R^n.$

A more detailed construction is needed for the topology of this space because the space $C\_0^\{\backslash infty\}(U)$ is not complete in the uniform norm. The topology on $D(U)$ is defined as follows: for any fixed compact set $K\; \backslash subseteq\; U,$ the space $C\_0^\{\backslash infty\}(K)$ of functions $f\; \backslash in\; C\_0^\{\backslash infty\}$ with $\backslash operatorname\{supp\}(f)\; \backslash subseteq\; K$ is a Fréchet space with countable family of seminorms $\backslash |f\backslash |\_m\; =\; \backslash sup\_\{k\; \backslash leq\; m\}\; \backslash sup\_x\; \backslash left|D^k\; f(x)\backslash right|$ (these are actually norms, and the completion of the space $C\_0^\{\backslash infty\}(K)$ with the $\backslash |\; \backslash cdot\; \backslash |\_m$ norm is a Banach space $D^m(K)$).

Given any collection $\backslash left(K\_a\backslash right)\_\{a\backslash in\; A\}$ of compact sets, directed by inclusion and such that their union equal $U,$ the $C\_0^\{\backslash infty\}\backslash left(K\_a\backslash right)$ form a direct system, and $D(U)$ is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, $D(U)$ is the union of all the $C\_0^\{\backslash infty\}\backslash left(K\_a\backslash right)$ with the strongest {{em|locally convex}} topology which makes each inclusion map $C\_0^\{\backslash infty\}\backslash left(K\_a\backslash right)\; \backslash hookrightarrow\; D(U)$ continuous.

This space is locally convex and complete. However, it is not metrizable, and so it is not a Fréchet space. The dual space of $D\backslash left(\backslash R^n\backslash right)$ is the space of distributions on $\backslash R^n.$

More abstractly, given a topological space $X,$ the space $C(X)$ of continuous (not necessarily bounded) functions on $X$ can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms $\backslash varphi\_K(f)\; =\; \backslash max\; \backslash \{|f(x)|\; :\; x\; \backslash in\; K\backslash \}$ (as $K$ varies over the directed set of all compact subsets of $X$). When $X$ is locally compact (for example, an open set in $\backslash R^n$) the Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of $C(X)$ that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.

Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:

• The spaces $L^p([0,\; 1])$ for are equipped with the F-norm $$\backslash |f\backslash |^p\_p\; =\; \backslash int\_0^1\; |f(x)|^p\; \backslash ,\; dx.$$ They are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces $L^p(\backslash mu)$ with an atomless, finite measure $\backslash mu$ and are not locally convex.
• The space of measurable functions on the unit interval $[0,\; 1]$ (where we identify two functions that are equal almost everywhere) has a vector-space topology defined by the translation-invariant metric (which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability): $$d(f,\; g)\; =\; \backslash int\_0^1\; \backslash frac$$

  f(x) - g(x)

  {1+|f(x) - g(x)|} \, dx. This space is often denoted $L\_0.$

Both examples have the property that any continuous linear map to the real numbers is $0.$ In particular, their dual space is trivial, that is, it contains only the zero functional.

• The sequence space $\backslash ell^p(\backslash N),$ is not locally convex.

{{Main|Continuous linear map}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=126–128}}|math_statement=

Let $T\; :\; X\; \backslash to\; Y$ be a linear operator between TVSs where $Y$ is locally convex (note that $X$ need {{em|not}} be locally convex). Then $T$ is continuous if and only if for every continuous seminorm $q$ on $Y$, there exists a continuous seminorm $p$ on $X$ such that $q\; \backslash circ\; T\; \backslash leq\; p.$

}}

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps.

Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces $X$ and $Y$ with families of seminorms $\backslash left(p\_\backslash alpha\backslash right)\_\{\backslash alpha\}$ and $\backslash left(q\_\backslash beta\backslash right)\_\{\backslash beta\}$ respectively, a linear map $T\; :\; X\; \backslash to\; Y$ is continuous if and only if for every $\backslash beta,$ there exist $\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n$ and $M\; >\; 0$ such that for all $v\; \backslash in\; X,$

$$q\_\backslash beta(Tv)\; \backslash leq\; M\; \backslash left(p\_\{\backslash alpha\_1\}(v)\; +\backslash dotsb+p\_\{\backslash alpha\_n\}(v)\backslash right).$$

In other words, each seminorm of the range of $T$ is bounded above by some finite sum of seminorms in the domain. If the family $\backslash left(p\_\backslash alpha\backslash right)\_\{\backslash alpha\}$ is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar:

$$q\_\backslash beta(Tv)\; \backslash leq\; Mp\_\backslash alpha(v).$$

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=126–128}}|math_statement=

If $X$ is a TVS (not necessarily locally convex) and if $f$ is a linear functional on $X$, then $f$ is continuous if and only if there exists a continuous seminorm $p$ on $X$ such that $|f|\; \backslash leq\; p.$

}}

If $X$ is a real or complex vector space, $f$ is a linear functional on $X$, and $p$ is a seminorm on $X$, then $|f|\; \backslash leq\; p$ if and only if $f\; \backslash leq\; p.${{sfn|Narici|Beckenstein|2011|pp=126-–128}}

If $f$ is a non-0 linear functional on a real vector space $X$ and if $p$ is a seminorm on $X$, then $f\; \backslash leq\; p$ if and only if {{sfn|Narici|Beckenstein|2011|pp=149–153}}

Let $n\; \backslash geq\; 1$ be an integer, $X\_1,\; \backslash ldots,\; X\_n$ be TVSs (not necessarily locally convex), let $Y$ be a locally convex TVS whose topology is determined by a family $\backslash mathcal\{Q\}$ of continuous seminorms, and let $M\; :\; \backslash prod\_\{i=1\}^n\; X\_i\; \backslash to\; Y$ be a multilinear operator that is linear in each of its $n$ coordinates.

The following are equivalent:

1. $M$ is continuous.
2. For every $q\; \backslash in\; \backslash mathcal\{Q\},$ there exist continuous seminorms $p\_1,\; \backslash ldots,\; p\_n$ on $X\_1,\; \backslash ldots,\; X\_n,$ respectively, such that $q(M(x))\; \backslash leq\; p\_1\backslash left(x\_1\backslash right)\; \backslash cdots\; p\_n\backslash left(x\_n\backslash right)$ for all $x\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)\; \backslash in\; \backslash prod\_\{i=1\}^n\; X\_i.${{sfn|Narici|Beckenstein|2011|pp=149–153}}
3. For every $q\; \backslash in\; \backslash mathcal\{Q\},$ there exists some neighborhood of the origin in $\backslash prod\_\{i=1\}^\{n\}\; X\_\{i\}$ on which $q\; \backslash circ\; M$ is bounded.{{sfn|Narici|Beckenstein|2011|pp=149–153}}

• {{annotated link|Convex metric space}}
• {{annotated link|Krein–Milman theorem}}
• {{annotated link|Linear form}}
• {{annotated link|Locally convex vector lattice}}
• {{annotated link|Minkowski functional}}
• {{annotated link|Seminorm}}
• {{annotated link|Sublinear functional}}
• {{annotated link|Topological group}}
• {{annotated link|Topological vector space}}
• {{annotated link|Vector space}}

{{reflist|group=note}}

{{reflist}}

{{reflist|group=proof}}

• {{Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition}}
• {{Berberian Lectures in Functional Analysis and Operator Theory}}
• {{citation|last1=Bessaga|first1=C.|last2=Pełczyński|first2=A.|title=Selected Topics in Infinite-Dimensional Topology|series=Monografie Matematyczne|publisher=Panstwowe wyd. naukowe|location=Warszawa|year=1975|url=https://books.google.com/books?id=7n9sAAAAMAAJ}}.
• {{Bourbaki Topological Vector Spaces}}
• {{Conway A Course in Functional Analysis}}
• {{cite book|last=Dunford|first=Nelson|title=Linear operators|publisher=Interscience Publishers|publication-place=New York|year=1988|isbn=0-471-60848-3|oclc=18412261|language=ro}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Grothendieck Topological Vector Spaces}}
• {{Jarchow Locally Convex Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces}}

{{Functional analysis}}

{{Topological vector spaces}}

{{Convex analysis and variational analysis}}

Category:Convex analysis

Category:Functional analysis

Category:Topological vector spaces