Auxiliary normed space

Throughout this article, $X$ will be a real or complex vector space (not necessarily a TVS, yet) and $D$ will be a disk in $X.$

Let $X$ will be a real or complex vector space. For any subset $D$ of $X,$ the Minkowski functional of $D$ defined by:

• If $D\; =\; \backslash varnothing$ then define $p\_\{\backslash varnothing\}(x)\; :\; \backslash \{0\backslash \}\; \backslash to\; [0,\; \backslash infty)$ to be the trivial map $p\_\{\backslash varnothing\}\; =\; 0${{sfn|Schaefer|Wolff|1999|p=169}} and it will be assumed that $\backslash operatorname\{span\}\; \backslash varnothing\; =\; \backslash \{0\backslash \}.$This is the smallest vector space containing $\backslash varnothing.$ Alternatively, if $D\; =\; \backslash varnothing$ then $D$ may instead be replaced with $\backslash \{0\backslash \}.$
• If $D\; \backslash neq\; \backslash varnothing$ and if $D$ is absorbing in $\backslash operatorname\{span\}\; D$ then denote the Minkowski functional of $D$ in $\backslash operatorname\{span\}\; D$ by $$p\_D\; :\; \backslash operatorname\{span\}\; D\; \backslash to\; [0,\; \backslash infty)$$ where for all $x\; \backslash in\; \backslash operatorname\{span\}\; D,$ this is defined by $$p\_D\; (x)\; :=\; \backslash inf\_\{\}\; \backslash \{r\; :\; x\; \backslash in\; r\; D,\; r\; >\; 0\backslash \}.$$

Let $X$ will be a real or complex vector space. For any subset $D$ of $X$ such that the Minkowski functional $p\_D$ is a seminorm on $\backslash operatorname\{span\}\; D,$ let $X\_D$ denote

$$X\_D\; :=\; \backslash left(\backslash operatorname\{span\}\; D,\; p\_D\backslash right)$$

which is called the seminormed space induced by $D,$ where if $p\_D$ is a norm then it is called the normed space induced by $D.$

Assumption (Topology): $X\_D\; =\; \backslash operatorname\{span\}\; D$ is endowed with the seminorm topology induced by $p\_D,$ which will be denoted by $\backslash tau\_D$ or $\backslash tau\_\{p\_D\}$

Importantly, this topology stems entirely from the set $D,$ the algebraic structure of $X,$ and the usual topology on $\backslash R$ (since $p\_D$ is defined using {{em|only}} the set $D$ and scalar multiplication). This justifies the study of Banach disks and is part of the reason why they play an important role in the theory of nuclear operators and nuclear spaces.

The inclusion map $\backslash operatorname\{In\}\_D\; :\; X\_D\; \backslash to\; X$ is called the canonical map.{{sfn|Schaefer|Wolff|1999|p=97}}

Suppose that $D$ is a disk.

Then $\backslash operatorname\{span\}\; D\; =\; \backslash bigcup\_\{n=1\}^\{\backslash infty\}\; n\; D$ so that $D$ is absorbing in $\backslash operatorname\{span\}\; D,$ the linear span of $D.$

The set $\backslash \{r\; D\; :\; r\; >\; 0\backslash \}$ of all positive scalar multiples of $D$ forms a basis of neighborhoods at the origin for a locally convex topological vector space topology $\backslash tau\_D$ on $\backslash operatorname\{span\}\; D.$

The Minkowski functional of the disk $D$ in $\backslash operatorname\{span\}\; D$ guarantees that $p\_D$ is well-defined and forms a seminorm on $\backslash operatorname\{span\}\; D.${{sfn|Trèves|2006|p=370}}

The locally convex topology induced by this seminorm is the topology $\backslash tau\_D$ that was defined before.

A bounded disk $D$ in a topological vector space $X$ such that $\backslash left(X\_D,\; p\_D\backslash right)$ is a Banach space is called a Banach disk, infracomplete, or a bounded completant in $X.$

If its shown that $\backslash left(\backslash operatorname\{span\}\; D,\; p\_D\backslash right)$ is a Banach space then $D$ will be a Banach disk in {{em|any}} TVS that contains $D$ as a bounded subset.

This is because the Minkowski functional $p\_D$ is defined in purely algebraic terms.

Consequently, the question of whether or not $\backslash left(X\_D,\; p\_D\backslash right)$ forms a Banach space is dependent only on the disk $D$ and the Minkowski functional $p\_D,$ and not on any particular TVS topology that $X$ may carry.

Thus the requirement that a Banach disk in a TVS $X$ be a bounded subset of $X$ is the only property that ties a Banach disk's topology to the topology of its containing TVS $X.$

Bounded disks

The following result explains why Banach disks are required to be bounded.

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=370-373}}{{sfn|Narici|Beckenstein|2011|pp=441-457}}{{sfn|Schaefer|Wolff|1999|p=97}} |math_statement=

If $D$ is a disk in a topological vector space (TVS) $X,$ then $D$ is bounded in $X$ if and only if the inclusion map $\backslash operatorname\{In\}\_D\; :\; X\_D\; \backslash to\; X$ is continuous.

}}

{{math proof|proof=

If the disk $D$ is bounded in the TVS $X$ then for all neighborhoods $U$ of the origin in $X,$ there exists some $r\; >\; 0$ such that $r\; D\; \backslash subseteq\; U\; \backslash cap\; X\_D.$

It follows that in this case the topology of $\backslash left(X\_D,\; p\_D\backslash right)$ is finer than the subspace topology that $X\_D$ inherits from $X,$ which implies that the inclusion map $\backslash operatorname\{In\}\_D\; :\; X\_D\; \backslash to\; X$ is continuous.

Conversely, if $X$ has a TVS topology such that $\backslash operatorname\{In\}\_D\; :\; X\_D\; \backslash to\; X$ is continuous, then for every neighborhood $U$ of the origin in $X$ there exists some $r\; >\; 0$ such that $r\; D\; \backslash subseteq\; U\; \backslash cap\; X\_D,$ which shows that $D$ is bounded in $X.$

}}

Hausdorffness

The space $\backslash left(X\_D,\; p\_D\backslash right)$ is Hausdorff if and only if $p\_D$ is a norm, which happens if and only if $D$ does not contain any non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|pp=115-154}}

In particular, if there exists a Hausdorff TVS topology on $X$ such that $D$ is bounded in $X$ then $p\_D$ is a norm.

An example where $X\_D$ is not Hausdorff is obtained by letting $X\; =\; \backslash R^2$ and letting $D$ be the $x$-axis.

Convergence of nets

Suppose that $D$ is a disk in $X$ such that $X\_D$ is Hausdorff and let $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ be a net in $X\_D.$

Then $x\_\backslash bull\; \backslash to\; 0$ in $X\_D$ if and only if there exists a net $r\_\backslash bull\; =\; \backslash left(r\_i\backslash right)\_\{i\; \backslash in\; I\}$ of real numbers such that $r\_\backslash bull\; \backslash to\; 0$ and $x\_i\; \backslash in\; r\_i\; D$ for all $i$;

moreover, in this case it will be assumed without loss of generality that $r\_i\; \backslash geq\; 0$ for all $i.$

Relationship between disk-induced spaces

If $C\; \backslash subseteq\; D\; \backslash subseteq\; X$ then $\backslash operatorname\{span\}\; C\; \backslash subseteq\; \backslash operatorname\{span\}\; D$ and $p\_D\; \backslash leq\; p\_C$ on $\backslash operatorname\{span\}\; C,$ so define the following continuous{{sfn|Narici|Beckenstein|2011|pp=441-457}} linear map:

If $C$ and $D$ are disks in $X$ with $C\; \backslash subseteq\; D$ then call the inclusion map $\backslash operatorname\{In\}\_C^D\; :\; X\_C\; \backslash to\; X\_D$ the canonical inclusion of $X\_C$ into $X\_D.$

In particular, the subspace topology that $\backslash operatorname\{span\}\; C$ inherits from $\backslash left(X\_D,\; p\_D\backslash right)$ is weaker than $\backslash left(X\_C,\; p\_C\backslash right)$'s seminorm topology.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

The disk as the closed unit ball

The disk $D$ is a closed subset of $\backslash left(X\_D,\; p\_D\backslash right)$ if and only if $D$ is the closed unit ball of the seminorm $p\_D$; that is,

$D\; =\; \backslash left\backslash \{x\; \backslash in\; \backslash operatorname\{span\}\; D\; :\; p\_D(x)\; \backslash leq\; 1\backslash right\backslash \}.$

If $D$ is a disk in a vector space $X$ and if there exists a TVS topology $\backslash tau$ on $\backslash operatorname\{span\}\; D$ such that $D$ is a closed and bounded subset of $\backslash left(\backslash operatorname\{span\}\; D,\; \backslash tau\backslash right),$ then $D$ is the closed unit ball of $\backslash left(X\_D,\; p\_D\backslash right)$ (that is, $D\; =\; \backslash left\backslash \{x\; \backslash in\; \backslash operatorname\{span\}\; D\; :\; p\_D(x)\; \backslash leq\; 1\backslash right\backslash \}$ ) (see footnote for proof).Assume WLOG that $X\; =\; \backslash operatorname\{span\}\; D.$ Since $D$ is closed in $(X,\; \backslash tau),$ it is also closed in $\backslash left(X\_D,\; p\_D\backslash right)$ and since the seminorm $p\_D$ is the Minkowski functional of $D,$ which is continuous on $\backslash left(X\_D,\; p\_D\backslash right),$ it follows {{harvtxt| Narici|Beckenstein| 2011|pp=119–120}} that $D$ is the closed unit ball in $\backslash left(X\_D,\; p\backslash right).$

The following theorem may be used to establish that $\backslash left(X\_D,\; p\_D\backslash right)$ is a Banach space.

Once this is established, $D$ will be a Banach disk in any TVS in which $D$ is bounded.

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=441-442}}|math_statement=

Let $D$ be a disk in a vector space $X.$

If there exists a Hausdorff TVS topology $\backslash tau$ on $\backslash operatorname\{span\}\; D$ such that $D$ is a bounded sequentially complete subset of $(\backslash operatorname\{span\}\; D,\; \backslash tau),$ then $\backslash left(X\_D,\; p\_D\backslash right)$ is a Banach space.

}}

{{math proof|proof=

Assume without loss of generality that $X\; =\; \backslash operatorname\{span\}\; D$ and let $p\; :=\; p\_D$ be the Minkowski functional of $D.$

Since $D$ is a bounded subset of a Hausdorff TVS, $D$ do not contain any non-trivial vector subspace, which implies that $p$ is a norm.

Let $\backslash tau\_D$ denote the norm topology on $X$ induced by $p$ where since $D$ is a bounded subset of $(X,\; \backslash tau),$ $\backslash tau\_D$ is finer than $\backslash tau.$

Because $D$ is convex and balanced, for any

$$2^\{-(n+1)\}\; D\; +\; \backslash cdots\; +\; 2^\{-(m+2)\}\; D\; =\; 2^\{-(m+1)\}\; \backslash left(1\; -\; 2^\{m-n\}\backslash right)\; D\; \backslash subseteq\; 2^\{-(m+2)\}\; D.$$

Let $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ be a Cauchy sequence in $\backslash left(X\_D,\; p\backslash right).$

By replacing $x\_\{\backslash bull\}$ with a subsequence, we may assume without loss of generality^† that for all $i,$

$$x\_\{i+1\}\; -\; x\_i\; \backslash in\; \backslash frac\{1\}\{2^\{i+2\}\}\; D.$$

This implies that for any

$$x\_n\; -\; x\_m\; =\; \backslash left(x\_n\; -\; x\_\{n-1\}\backslash right)\; +\; \backslash left(x\_\{m+1\}\; -\; x\_m\backslash right)\; \backslash in\; 2^\{-(n+1)\}\; D\; +\; \backslash cdots\; +\; 2^\{-(m+2)\}\; D\; \backslash subseteq\; 2^\{-(m+2)\}\; D$$

so that in particular, by taking $m\; =\; 1$ it follows that $x\_\{\backslash bull\}$ is contained in $x\_1\; +\; 2^\{-3\}\; D.$

Since $\backslash tau\_D$ is finer than $\backslash tau,$ $x\_\{\backslash bull\}$ is a Cauchy sequence in $(X,\; \backslash tau).$

For all $m\; >\; 0,$ $2^\{-(m+2)\}\; D$ is a Hausdorff sequentially complete subset of $(X,\; \backslash tau).$

In particular, this is true for $x\_1\; +\; 2^\{-3\}\; D$ so there exists some $x\; \backslash in\; x\_1\; +\; 2^\{-3\}\; D$ such that $x\_\{\backslash bull\}\; \backslash to\; x$ in $(X,\; \backslash tau).$

Since $x\_n\; -\; x\_m\; \backslash in\; 2^\{-(m+2)\}\; D$ for all by fixing $m$ and taking the limit (in $(X,\; \backslash tau)$) as $n\; \backslash to\; \backslash infty,$ it follows that $x\; -\; x\_m\; \backslash in\; 2^\{-(m+2)\}\; D$ for each $m\; >\; 0.$

This implies that $p\backslash left(x\; -\; x\_m\backslash right)\; \backslash to\; 0$ as $m\; \backslash to\; \backslash infty,$ which says exactly that $x\_\{\backslash bull\}\; \backslash to\; x$ in $\backslash left(X\_D,\; p\backslash right).$

This shows that $\backslash left(X\_D,\; p\backslash right)$ is complete.

^†This assumption is allowed because $x\_\{\backslash bull\}$ is a Cauchy sequence in a metric space (so the limits of all subsequences are equal) and a sequence in a metric space converges if and only if every subsequence has a sub-subsequence that converges.

}}

Note that even if $D$ is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that $\backslash left(X\_D,\; p\_D\backslash right)$ is a Banach space by applying this theorem to some disk $K$ satisfying

because $p\_D\; =\; p\_K.$

The following are consequences of the above theorem:

• A sequentially complete bounded disk in a Hausdorff TVS is a Banach disk.{{sfn|Narici|Beckenstein|2011|pp=441-457}}
• Any disk in a Hausdorff TVS that is complete and bounded (e.g. compact) is a Banach disk.{{sfn|Trèves|2006|pp=370–371}}
• The closed unit ball in a Fréchet space is sequentially complete and thus a Banach disk.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

Suppose that $D$ is a bounded disk in a TVS $X.$

• If $L\; :\; X\; \backslash to\; Y$ is a continuous linear map and $B\; \backslash subseteq\; X$ is a Banach disk, then $L(B)$ is a Banach disk and $L\backslash big\backslash vert\_\{X\_B\}\; :\; X\_B\; \backslash to\; L\backslash left(X\_B\backslash right)$ induces an isometric TVS-isomorphism $Y\_\{L(B)\}\; \backslash cong\; X\_B\; /\; \backslash left(X\_B\; \backslash cap\; \backslash operatorname\{ker\}\; L\backslash right).$

Let $X$ be a TVS and let $D$ be a bounded disk in $X.$

If $D$ is a bounded Banach disk in a Hausdorff locally convex space $X$ and if $T$ is a barrel in $X$ then $T$ absorbs $D$ (that is, there is a number $r\; >\; 0$ such that $D\; \backslash subseteq\; r\; T.${{sfn|Trèves|2006|pp=370-373}}

If $U$ is a convex balanced closed neighborhood of the origin in $X$ then the collection of all neighborhoods $r\; U,$ where $r\; >\; 0$ ranges over the positive real numbers, induces a topological vector space topology on $X.$ When $X$ has this topology, it is denoted by $X\_U.$ Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space $X\; /\; p\_U^\{-1\}(0)$ is denoted by $\backslash overline\{X\_U\}$ so that $\backslash overline\{X\_U\}$ is a complete Hausdorff space and $p\_U(x)\; :=\; \backslash inf\_\{x\; \backslash in\; r\; U,\; r\; >\; 0\}\; r$ is a norm on this space making $\backslash overline\{X\_U\}$ into a Banach space. The polar of $U,$ $U^\{\backslash circ\},$ is a weakly compact bounded equicontinuous disk in $X^\{\backslash prime\}$ and so is infracomplete.

If $X$ is a metrizable locally convex TVS then for every bounded subset $B$ of $X,$ there exists a bounded disk $D$ in $X$ such that $B\; \backslash subseteq\; X\_D,$ and both $X$ and $X\_D$ induce the same subspace topology on $B.${{sfn|Narici|Beckenstein|2011|pp=441-457}}

Suppose that $X$ is a topological vector space and $V$ is a convex balanced and radial set.

Then $\backslash left\backslash \{\backslash tfrac\{1\}\{n\}\; V\; :\; n\; =\; 1,\; 2,\; \backslash ldots\backslash right\backslash \}$ is a neighborhood basis at the origin for some locally convex topology $\backslash tau\_V$ on $X.$

This TVS topology $\backslash tau\_V$ is given by the Minkowski functional formed by $V,$ $p\_V\; :\; X\; \backslash to\; \backslash R,$ which is a seminorm on $X$ defined by $p\_V(x)\; :=\; \backslash inf\_\{x\; \backslash in\; r\; V,\; r\; >\; 0\}\; r.$

The topology $\backslash tau\_V$ is Hausdorff if and only if $p\_V$ is a norm, or equivalently, if and only if $X\; /\; p\_V^\{-1\}(0)\; =\; \backslash \{0\backslash \}$ or equivalently, for which it suffices that $V$ be bounded in $X.$

The topology $\backslash tau\_V$ need not be Hausdorff but $X\; /\; p\_V^\{-1\}(0)$ is Hausdorff.

A norm on $X\; /\; p\_V^\{-1\}(0)$ is given by $\backslash left\backslash |x\; +\; X\; /\; p\_V^\{-1\}(0)\backslash right\backslash |\; :=\; p\_V(x),$ where this value is in fact independent of the representative of the equivalence class $x\; +\; X\; /\; p\_V^\{-1\}(0)$ chosen.

The normed space $\backslash left(X\; /\; p\_V^\{-1\}(0),\; \backslash |\; \backslash cdot\; \backslash |\backslash right)$ is denoted by $X\_V$ and its completion is denoted by $\backslash overline\{X\_V\}.$

If in addition $V$ is bounded in $X$ then the seminorm $p\_V\; :\; X\; \backslash to\; \backslash R$ is a norm so in particular, $p\_V^\{-1\}(0)\; =\; \backslash \{0\backslash \}.$

In this case, we take $X\_V$ to be the vector space $X$ instead of $X\; /\; \backslash \{0\backslash \}$ so that the notation $X\_V$ is unambiguous (whether $X\_V$ denotes the space induced by a radial disk or the space induced by a bounded disk).{{sfn|Schaefer|Wolff|1999|p=97}}

The quotient topology $\backslash tau\_Q$ on $X\; /\; p\_V^\{-1\}(0)$ (inherited from $X$'s original topology) is finer (in general, strictly finer) than the norm topology.

The canonical map is the quotient map $q\_V\; :\; X\; \backslash to\; X\_V\; =\; X\; /\; p\_V^\{-1\}(0),$ which is continuous when $X\_V$ has either the norm topology or the quotient topology.{{sfn|Schaefer|Wolff|1999|p=97}}

If $U$ and $V$ are radial disks such that $U\; \backslash subseteq\; V$ then $p\_U^\{-1\}(0)\; \backslash subseteq\; p\_V^\{-1\}(0)$ so there is a continuous linear surjective canonical map $q\_\{V,U\}\; :\; X\; /\; p\_U^\{-1\}(0)\; \backslash to\; X\; /\; p\_V^\{-1\}(0)\; =\; X\_V$ defined by sending

$x\; +\; p\_U^\{-1\}(0)\; \backslash in\; X\_U\; =\; X\; /\; p\_U^\{-1\}(0)$ to the equivalence class $x\; +\; p\_V^\{-1\}(0),$ where one may verify that the definition does not depend on the representative of the equivalence class $x\; +\; p\_U^\{-1\}(0)$ that is chosen.{{sfn|Schaefer|Wolff|1999|p=97}}

This canonical map has norm $\backslash ,\backslash leq\; 1${{sfn|Schaefer|Wolff|1999|p=97}} and it has a unique continuous linear canonical extension to $\backslash overline\{X\_U\}$ that is denoted by $\backslash overline\{g\_\{V,U\}\}\; :\; \backslash overline\{X\_U\}\; \backslash to\; \backslash overline\{X\_V\}.$

Suppose that in addition $B\; \backslash neq\; \backslash varnothing$ and $C$ are bounded disks in $X$ with $B\; \backslash subseteq\; C$ so that $X\_B\; \backslash subseteq\; X\_C$ and the inclusion $\backslash operatorname\{In\}\_B^C\; :\; X\_B\; \backslash to\; X\_C$ is a continuous linear map.

Let $\backslash operatorname\{In\}\_B\; :\; X\_B\; \backslash to\; X,$ $\backslash operatorname\{In\}\_C\; :\; X\_C\; \backslash to\; X,$ and $\backslash operatorname\{In\}\_B^C\; :\; X\_B\; \backslash to\; X\_C$ be the canonical maps.

Then $\backslash operatorname\{In\}\_C\; =\; \backslash operatorname\{In\}\_B^C\; \backslash circ\; \backslash operatorname\{In\}\_C\; :\; X\_B\; \backslash to\; X\_C$ and $q\_V\; =\; q\_\{V,U\}\; \backslash circ\; q\_U.${{sfn|Schaefer|Wolff|1999|p=97}}

Suppose that $S$ is a bounded radial disk.

Since $S$ is a bounded disk, if $D\; :=\; S$ then we may create the auxiliary normed space $X\_D\; =\; \backslash operatorname\{span\}\; D$ with norm $p\_D(x)\; :=\; \backslash inf\_\{x\; \backslash in\; r\; D,\; r\; >\; 0\}\; r$; since $S$ is radial, $X\_S\; =\; X.$

Since $S$ is a radial disk, if $V\; :=\; S$ then we may create the auxiliary seminormed space $X\; /\; p\_V^\{-1\}(0)$ with the seminorm $p\_V(x)\; :=\; \backslash inf\_\{x\; \backslash in\; r\; V,\; r\; >\; 0\}\; r$; because $S$ is bounded, this seminorm is a norm and $p\_V^\{-1\}(0)\; =\; \backslash \{0\backslash \}$ so $X\; /\; p\_V^\{-1\}(0)\; =\; X\; /\; \backslash \{0\backslash \}\; =\; X.$

Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.

Suppose that $H$ is a weakly closed equicontinuous disk in $X^\{\backslash prime\}$ (this implies that $H$ is weakly compact) and let

$$U\; :=\; H^\{\backslash circ\}\; =\; \backslash \{x\; \backslash in\; X\; :\; |h(x)|\; \backslash leq\; 1\; \backslash text\{\; for\; all\; \}\; h\; \backslash in\; H\backslash \}$$

be the polar of $H.$

Because $U^\backslash circ\; =\; H^\{\backslash circ\backslash circ\}\; =\; H$ by the bipolar theorem, it follows that a continuous linear functional $f$ belongs to $X^\{\backslash prime\}\_H\; =\; \backslash operatorname\{span\}\; H$ if and only if $f$ belongs to the continuous dual space of $\backslash left(X,\; p\_U\backslash right),$ where $p\_U$ is the Minkowski functional of $U$ defined by $p\_U(x)\; :=\; \backslash inf\_\{x\; \backslash in\; r\; U,\; r\; >\; 0\}\; r.${{sfn|Trèves|2006|p=477}}

A disk in a TVS is called infrabornivorous{{sfn|Narici|Beckenstein|2011|pp=441-457}} if it absorbs all Banach disks.

A linear map between two TVSs is called infrabounded{{sfn|Narici|Beckenstein|2011|pp=441-457}} if it maps Banach disks to bounded disks.

A sequence $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty$ in a TVS $X$ is said to be fast convergent{{sfn|Narici|Beckenstein|2011|pp=441-457}} to a point $x\; \backslash in\; X$ if there exists a Banach disk $D$ such that both $x$ and the sequence is (eventually) contained in $\backslash operatorname\{span\}\; D$ and $x\_\backslash bull\; \backslash to\; x$ in $\backslash left(X\_D,\; p\_D\backslash right).$

Every fast convergent sequence is Mackey convergent.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

• {{annotated link|Bornological space}}
• {{annotated link|Injective tensor product}}
• {{annotated link|Locally convex topological vector space}}
• {{annotated link|Nuclear operator}}
• {{annotated link|Nuclear space}}
• {{annotated link|Initial topology}}
• {{annotated link|Projective tensor product}}
• {{annotated link|Schwartz topological vector space}}
• {{annotated link|Tensor product of Hilbert spaces}}
• {{annotated link|Topological tensor product}}
• {{annotated link|Ultrabornological space}}

{{reflist|group=note}}

{{reflist}}

• {{cite journal|last1=Burzyk|first1=Józef|last2=Gilsdorf|first2=Thomas E.|title=Some remarks about Mackey convergence|journal=International Journal of Mathematics and Mathematical Sciences|publisher=Hindawi Limited|volume=18|issue=4|year=1995|issn=0161-1712|doi=10.1155/s0161171295000846|url=https://downloads.hindawi.com/journals/ijmms/1995/138625.pdf|pages=659–664 |doi-access=free }}
• {{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}}
• {{Dubinsky The Structure of Nuclear Fréchet Spaces}}
• {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}}
• {{Hogbe-Nlend Bornologies and Functional Analysis}}
• {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}}
• {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Pietsch Nuclear Locally Convex Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Ryan Introduction to Tensor Products of Banach Spaces|edition=1}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}}

• [https://ncatlab.org/nlab/show/nuclear+space Nuclear space at ncatlab]

{{Functional analysis}}

{{TopologicalTensorProductsAndNuclearSpaces}}

Category:Functional analysis