View source for Auxiliary normed space

In [[Functional analysis|functional analysis]], two methods of constructing [[Normed vector space|normed spaces]] from [[Absolutely convex set|disk]]s were systematically employed by [[Biography:Alexander Grothendieck|Alexander Grothendieck]] to define [[Nuclear operator|nuclear operator]]s and [[Nuclear space|nuclear space]]s.{{sfn|Schaefer|Wolff|1999|p=97}} 
One method is used if the disk <math>D</math> is bounded: in this case, the '''auxiliary normed space''' is <math>\operatorname{span} D</math> with norm 
<math display="block">p_D(x) := \inf_{x \in r D, r > 0} r.</math> 
The other method is used if the disk <math>D</math> is [[Absorbing set|absorbing]]: in this case, the auxiliary normed space is the [[Quotient space (linear algebra)|quotient space]] <math>X / p_D^{-1}(0).</math> 
If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces).

==Induced by a bounded disk – Banach disks==

Throughout this article, <math>X</math> will be a real or complex vector space (not necessarily a TVS, yet) and <math>D</math> will be a [[Absolutely convex set|disk]] in <math>X.</math>

===Seminormed space induced by a disk===

Let <math>X</math> will be a real or complex vector space. For any subset <math>D</math> of <math>X,</math> the ''[[Minkowski functional]]'' of <math>D</math> defined by:
*If <math>D = \varnothing</math> then define <math>p_{\varnothing}(x) : \{0\} \to [0, \infty)</math> to be the trivial map <math>p_{\varnothing} = 0</math>{{sfn|Schaefer|Wolff|1999|p=169}} and it will be assumed that <math>\operatorname{span} \varnothing = \{0\}.</math><ref group=note>This is the smallest vector space containing <math>\varnothing.</math> Alternatively, if <math>D = \varnothing</math> then <math>D</math> may instead be replaced with <math>\{0\}.</math></ref>
*If <math>D \neq \varnothing</math> and if <math>D</math> is [[Absorbing set|absorbing]] in <math>\operatorname{span} D</math> then denote the [[Minkowski functional]] of <math>D</math> in <math>\operatorname{span} D</math> by <math display="block">p_D : \operatorname{span} D \to [0, \infty)</math> where for all <math>x \in \operatorname{span} D,</math> this is defined by <math display="block">p_D (x) := \inf_{} \{r : x \in r D, r > 0\}.</math>

Let <math>X</math> will be a real or complex vector space. For any subset <math>D</math> of <math>X</math> such that the Minkowski functional <math>p_D</math>is a [[Seminorm|seminorm]] on <math>\operatorname{span} D,</math> let <math>X_D</math> denote 
<math display="block">X_D := \left(\operatorname{span} D, p_D\right)</math>
which is called the ''seminormed space induced by <math>D,</math>'' where if <math>p_D</math> is a [[Norm (mathematics)|norm]] then it is called the ''normed space induced by <math>D.</math>''

'''Assumption''' ('''Topology'''): <math>X_D = \operatorname{span} D</math> is endowed with the seminorm topology induced by <math>p_D,</math> which will be denoted by <math>\tau_D</math> or <math>\tau_{p_D}</math>

Importantly, this topology stems ''entirely'' from the set <math>D,</math> the algebraic structure of <math>X,</math> and the usual topology on <math>\R</math> (since <math>p_D</math>is defined using {{em|only}} the set <math>D</math> 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 operator|nuclear operator]]s and [[Nuclear space|nuclear space]]s.

The inclusion map <math>\operatorname{In}_D : X_D \to X</math> is called the ''canonical map''.{{sfn|Schaefer|Wolff|1999|p=97}}

Suppose that <math>D</math> is a disk. 
Then <math display="inline"> \operatorname{span} D = \bigcup_{n=1}^{\infty} n D</math> so that <math>D</math> is [[Absorbing set|absorbing]] in <math>\operatorname{span} D,</math> the [[Linear span|linear span]] of <math>D.</math> 
The set <math>\{r D : r > 0\}</math> of all positive scalar multiples of <math>D</math> forms a basis of neighborhoods at the origin for a [[Locally convex topological vector space|locally convex topological vector space]] topology <math>\tau_D</math> on <math>\operatorname{span} D.</math> 
The [[Minkowski functional]] of the disk <math>D</math> in <math>\operatorname{span} D</math> guarantees that <math>p_D</math>is well-defined and forms a [[Seminorm|seminorm]] on <math>\operatorname{span} D.</math>{{sfn|Trèves|2006|p=370}} 
The locally convex topology induced by this seminorm is the topology <math>\tau_D</math> that was defined before.

===Banach disk definition===

A [[Bounded set (topological vector space)|bounded]] [[Absolutely convex set|disk]] <math>D</math> in a [[Topological vector space|topological vector space]] <math>X</math> such that <math>\left(X_D, p_D\right)</math> is a [[Banach space]] is called a '''Banach disk''', '''infracomplete''', or a '''bounded completant''' in <math>X.</math>

If its shown that <math>\left(\operatorname{span} D, p_D\right)</math> is a Banach space then <math>D</math> will be a Banach disk in {{em|any}} TVS that contains <math>D</math> as a bounded subset.

This is because the Minkowski functional <math>p_D</math>is defined in purely algebraic terms. 
Consequently, the question of whether or not <math>\left(X_D, p_D\right)</math> forms a Banach space is dependent only on the disk <math>D</math> and the Minkowski functional <math>p_D,</math> and not on any particular TVS topology that <math>X</math> may carry. 
Thus the requirement that a Banach disk in a TVS <math>X</math> be a bounded subset of <math>X</math> is the only property that ties a Banach disk's topology to the topology of its containing TVS <math>X.</math>

===Properties of disk induced seminormed spaces===

'''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 <math>D</math> is a disk in a [[Topological vector space|topological vector space]] (TVS) <math>X,</math> then <math>D</math> is [[Bounded set (topological vector space)|bounded]] in <math>X</math> if and only if the inclusion map <math>\operatorname{In}_D : X_D \to X</math> is continuous.
}}
{{math proof|proof=
If the disk <math>D</math> is bounded in the TVS <math>X</math> then for all neighborhoods <math>U</math> of the origin in <math>X,</math> there exists some <math>r > 0</math> such that <math>r D \subseteq U \cap X_D.</math> 
It follows that in this case the topology of <math>\left(X_D, p_D\right)</math> is finer than the subspace topology that <math>X_D</math> inherits from <math>X,</math> which implies that the inclusion map <math>\operatorname{In}_D : X_D \to X</math> is continuous. 
<!--<math>X_D</math>'s topology is in fact finer than even the subspace topology that <math>\operatorname{span} D</math> inherits from <math>X_{\tau},</math> which is <math>X</math> with the [[Mackey topology]].-->
Conversely, if <math>X</math> has a TVS topology such that <math>\operatorname{In}_D : X_D \to X</math> is continuous, then for every neighborhood <math>U</math> of the origin in <math>X</math> there exists some <math>r > 0</math> such that <math>r D \subseteq U \cap X_D,</math> which shows that <math>D</math> is bounded in <math>X.</math> 
}}

'''Hausdorffness'''

The space <math>\left(X_D, p_D\right)</math> is [[Hausdorff space|Hausdorff]] if and only if <math>p_D</math>is a norm, which happens if and only if <math>D</math> 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 <math>X</math> such that <math>D</math> is bounded in <math>X</math> then <math>p_D</math>is a norm. 
An example where <math>X_D</math> is not Hausdorff is obtained by letting <math>X = \R^2</math> and letting <math>D</math> be the <math>x</math>-axis.

'''Convergence of nets'''

Suppose that <math>D</math> is a disk in <math>X</math> such that <math>X_D</math> is Hausdorff and let <math>x_\bull = \left(x_i\right)_{i \in I}</math> be a net in <math>X_D.</math> 
Then <math>x_\bull \to 0</math> in <math>X_D</math> if and only if there exists a net <math>r_\bull = \left(r_i\right)_{i \in I}</math> of real numbers such that <math>r_\bull \to 0</math> and <math>x_i \in r_i D</math> for all <math>i</math>; 
moreover, in this case it will be assumed without loss of generality that <math>r_i \geq 0</math> for all <math>i.</math>

'''Relationship between disk-induced spaces'''

If <math>C \subseteq D \subseteq X</math>then <math>\operatorname{span} C \subseteq \operatorname{span} D</math> and <math>p_D \leq p_C</math> on <math>\operatorname{span} C,</math> so define the following continuous{{sfn|Narici|Beckenstein|2011|pp=441-457}} linear map:

If <math>C</math> and <math>D</math> are disks in <math>X</math> with <math>C \subseteq D</math> then call the inclusion map <math>\operatorname{In}_C^D : X_C \to X_D</math> the ''canonical inclusion'' of <math>X_C</math> into <math>X_D.</math>

In particular, the subspace topology that <math>\operatorname{span} C</math> inherits from <math>\left(X_D, p_D\right)</math> is weaker than <math>\left(X_C, p_C\right)</math>'s seminorm topology.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

'''The disk as the closed unit ball'''

The disk <math>D</math> is a closed subset of <math>\left(X_D, p_D\right)</math> if and only if <math>D</math> is the closed unit ball of the seminorm <math>p_D</math>; that is, 
<math>D = \left\{x \in \operatorname{span} D : p_D(x) \leq 1\right\}.</math> 

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

===Sufficient conditions for a Banach disk===

The following theorem may be used to establish that <math>\left(X_D, p_D\right)</math> is a Banach space. 
Once this is established, <math>D</math> will be a Banach disk in any TVS in which <math>D</math> is bounded.

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=441-442}}|math_statement= 
Let <math>D</math> be a disk in a vector space <math>X.</math> 
If there exists a Hausdorff TVS topology <math>\tau</math> on <math>\operatorname{span} D</math> such that <math>D</math> is a bounded [[Sequentially complete|sequentially complete]] subset of <math>(\operatorname{span} D, \tau),</math> then <math>\left(X_D, p_D\right)</math> is a Banach space. 
}}

{{math proof|proof=
Assume without loss of generality that <math>X = \operatorname{span} D</math> and let <math>p := p_D</math> be the [[Minkowski functional]] of <math>D.</math> 
Since <math>D</math> is a bounded subset of a Hausdorff TVS, <math>D</math> do not contain any non-trivial vector subspace, which implies that <math>p</math> is a norm. 
Let <math>\tau_D</math> denote the norm topology on <math>X</math> induced by <math>p</math> where since <math>D</math> is a bounded subset of <math>(X, \tau),</math> <math>\tau_D</math> is finer than <math>\tau.</math>

Because <math>D</math> is convex and balanced, for any <math>0 < m < n</math>

<math display="block">2^{-(n+1)} D + \cdots + 2^{-(m+2)} D = 2^{-(m+1)} \left(1 - 2^{m-n}\right) D \subseteq 2^{-(m+2)} D.</math>

Let <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> be a Cauchy sequence in <math>\left(X_D, p\right).</math> 
By replacing <math>x_{\bull}</math> with a subsequence, we may assume without loss of generality<sup>†</sup> that for all <math>i,</math>
<math display="block">x_{i+1} - x_i \in \frac{1}{2^{i+2}} D.</math>

This implies that for any <math>0 < m < n,</math>
<math display="block">x_n - x_m = \left(x_n - x_{n-1}\right) + \left(x_{m+1} - x_m\right) \in 2^{-(n+1)} D + \cdots + 2^{-(m+2)} D \subseteq 2^{-(m+2)} D</math>
so that in particular, by taking <math>m = 1</math> it follows that <math>x_{\bull}</math> is contained in <math>x_1 + 2^{-3} D.</math> 
Since <math>\tau_D</math> is finer than <math>\tau,</math> <math>x_{\bull}</math> is a Cauchy sequence in <math>(X, \tau).</math> 
For all <math>m > 0,</math> <math>2^{-(m+2)} D</math> is a Hausdorff sequentially complete subset of <math>(X, \tau).</math> 
In particular, this is true for <math>x_1 + 2^{-3} D</math> so there exists some <math>x \in x_1 + 2^{-3} D</math> such that <math>x_{\bull} \to x</math> in <math>(X, \tau).</math>

Since <math>x_n - x_m \in 2^{-(m+2)} D</math> for all <math>0 < m < n,</math> by fixing <math>m</math> and taking the limit (in <math>(X, \tau)</math>) as <math>n \to \infty,</math> it follows that <math>x - x_m \in 2^{-(m+2)} D</math> for each <math>m > 0.</math> 
This implies that <math>p\left(x - x_m\right) \to 0</math> as <math>m \to \infty,</math> which says exactly that <math>x_{\bull} \to x</math> in <math>\left(X_D, p\right).</math> 
This shows that <math>\left(X_D, p\right)</math> is complete.

<sup>†</sup>This assumption is allowed because <math>x_{\bull}</math> 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 <math>D</math> is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that <math>\left(X_D, p_D\right)</math> is a Banach space by applying this theorem to some disk <math>K</math> satisfying 
<math display="block">\left\{x \in \operatorname{span} D : p_D(x) < 1\right\} \subseteq K \subseteq \left\{x \in \operatorname{span} D : p_D(x) \leq 1\right\}</math>
because <math>p_D = p_K.</math>

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 <math>D</math> is a bounded disk in a TVS <math>X.</math>

*If <math>L : X \to Y</math> is a continuous linear map and <math>B \subseteq X</math> is a Banach disk, then <math>L(B)</math> is a Banach disk and <math>L\big\vert_{X_B} : X_B \to L\left(X_B\right)</math> induces an isometric TVS-isomorphism <math>Y_{L(B)} \cong X_B / \left(X_B \cap \operatorname{ker} L\right).</math>

===Properties of Banach disks===

Let <math>X</math> be a TVS and let <math>D</math> be a bounded disk in <math>X.</math>

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

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

If <math>X</math> is a [[Metrizable topological vector space|metrizable]] [[Locally convex topological vector space|locally convex]] TVS then for every [[Bounded set (topological vector space)|bounded]] subset <math>B</math> of <math>X,</math> there exists a bounded [[Absolutely convex set|disk]] <math>D</math> in <math>X</math> such that <math>B \subseteq X_D,</math> and both <math>X</math> and <math>X_D</math> induce the same [[Subspace topology|subspace topology]] on <math>B.</math>{{sfn|Narici|Beckenstein|2011|pp=441-457}}

==Induced by a radial disk – quotient==

Suppose that <math>X</math> is a topological vector space and <math>V</math> is a [[Convex set|convex]] [[Balanced set|balanced]] and [[Radial set|radial]] set. 
Then <math>\left\{\tfrac{1}{n} V : n = 1, 2, \ldots\right\}</math> is a neighborhood basis at the origin for some locally convex topology <math>\tau_V</math> on <math>X.</math> 
This TVS topology <math>\tau_V</math> is given by the [[Minkowski functional]] formed by <math>V,</math> <math>p_V : X \to \R,</math> which is a seminorm on <math>X</math> defined by <math>p_V(x) := \inf_{x \in r V, r > 0} r.</math> 
The topology <math>\tau_V</math> is Hausdorff if and only if <math>p_V</math> is a norm, or equivalently, if and only if <math>X / p_V^{-1}(0) = \{0\}</math> or equivalently, for which it suffices that <math>V</math> be bounded in <math>X.</math> 
The topology <math>\tau_V</math> need not be Hausdorff but <math>X / p_V^{-1}(0)</math> is Hausdorff. 
A norm on <math>X / p_V^{-1}(0)</math> is given by <math>\left\|x + X / p_V^{-1}(0)\right\| := p_V(x),</math> where this value is in fact independent of the representative of the equivalence class <math>x + X / p_V^{-1}(0)</math> chosen. 
The normed space <math>\left(X / p_V^{-1}(0), \| \cdot \|\right)</math> is denoted by <math>X_V</math> and its completion is denoted by <math>\overline{X_V}.</math>

If in addition <math>V</math> is bounded in <math>X</math> then the seminorm <math>p_V : X \to \R</math> is a norm so in particular, <math>p_V^{-1}(0) = \{0\}.</math> 
In this case, we take <math>X_V</math> to be the vector space <math>X</math> instead of <math>X / \{0\}</math> so that the notation <math>X_V</math> is unambiguous (whether <math>X_V</math> 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 <math>\tau_Q</math> on <math>X / p_V^{-1}(0)</math> (inherited from <math>X</math>'s original topology) is finer (in general, strictly finer) than the norm topology.

===Canonical maps===

The ''canonical map'' is the quotient map <math>q_V : X \to X_V = X / p_V^{-1}(0),</math> which is continuous when <math>X_V</math> has either the norm topology or the quotient topology.{{sfn|Schaefer|Wolff|1999|p=97}}

If <math>U</math> and <math>V</math> are radial disks such that <math>U \subseteq V</math>then <math>p_U^{-1}(0) \subseteq p_V^{-1}(0)</math> so there is a continuous linear surjective ''canonical map'' <math>q_{V,U} : X / p_U^{-1}(0) \to X / p_V^{-1}(0) = X_V</math> defined by sending 
<math>x + p_U^{-1}(0) \in X_U = X / p_U^{-1}(0)</math> to the equivalence class <math>x + p_V^{-1}(0),</math> where one may verify that the definition does not depend on the representative of the equivalence class <math>x + p_U^{-1}(0)</math> that is chosen.{{sfn|Schaefer|Wolff|1999|p=97}} 
This canonical map has norm <math>\,\leq 1</math>{{sfn|Schaefer|Wolff|1999|p=97}} and it has a unique continuous linear canonical extension to <math>\overline{X_U}</math> that is denoted by <math>\overline{g_{V,U}} : \overline{X_U} \to \overline{X_V}.</math>

Suppose that in addition <math>B \neq \varnothing</math> and <math>C</math> are bounded disks in <math>X</math> with <math>B \subseteq C</math> so that <math>X_B \subseteq X_C</math> and the inclusion <math>\operatorname{In}_B^C : X_B \to X_C</math> is a continuous linear map. 
Let <math>\operatorname{In}_B : X_B \to X,</math> <math>\operatorname{In}_C : X_C \to X,</math> and <math>\operatorname{In}_B^C : X_B \to X_C</math> be the canonical maps. 
Then <math>\operatorname{In}_C = \operatorname{In}_B^C \circ \operatorname{In}_C : X_B \to X_C</math> and <math>q_V = q_{V,U} \circ q_U.</math>{{sfn|Schaefer|Wolff|1999|p=97}}

==Induced by a bounded radial disk==

Suppose that <math>S</math> is a bounded radial disk. 
Since <math>S</math> is a bounded disk, if <math>D := S</math> then we may create the auxiliary normed space <math>X_D = \operatorname{span} D</math> with norm <math>p_D(x) := \inf_{x \in r D, r > 0} r</math>; since <math>S</math> is radial, <math>X_S = X.</math> 
Since <math>S</math> is a radial disk, if <math>V := S</math> then we may create the auxiliary seminormed space <math>X / p_V^{-1}(0)</math> with the seminorm <math>p_V(x) := \inf_{x \in r V, r > 0} r</math>; because <math>S</math> is bounded, this seminorm is a norm and <math>p_V^{-1}(0) = \{0\}</math> so <math>X / p_V^{-1}(0) = X / \{0\} = X.</math> 
Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.

==Duality==

Suppose that <math>H</math> is a weakly closed equicontinuous disk in <math>X^{\prime}</math> (this implies that <math>H</math> is weakly compact) and let
<math display="block">U := H^{\circ} = \{x \in X : |h(x)| \leq 1 \text{ for all } h \in H\}</math>
be the [[Polar set|polar]] of <math>H.</math> 
Because <math>U^\circ = H^{\circ\circ} = H</math> by the [[Bipolar theorem|bipolar theorem]], it follows that a continuous linear functional <math>f</math> belongs to <math>X^{\prime}_H = \operatorname{span} H</math> if and only if <math>f</math> belongs to the continuous dual space of <math>\left(X, p_U\right),</math> where <math>p_U</math> is the [[Minkowski functional]] of <math>U</math> defined by <math>p_U(x) := \inf_{x \in r U, r > 0} r.</math>{{sfn|Trèves|2006|p=477}}

==Related concepts==

A disk in a TVS is called ''infrabornivorous''{{sfn|Narici|Beckenstein|2011|pp=441-457}} if it [[Absorbing set|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.

===Fast convergence===

A sequence <math>x_\bull = \left(x_i\right)_{i=1}^\infty</math> in a TVS <math>X</math> is said to be ''fast convergent''{{sfn|Narici|Beckenstein|2011|pp=441-457}} to a point <math>x \in X</math> if there exists a Banach disk <math>D</math> such that both <math>x</math> and the sequence is (eventually) contained in <math>\operatorname{span} D</math> and <math>x_\bull \to x</math> in <math>\left(X_D, p_D\right).</math>

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

==See also==

* {{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}}

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

==Bibliography==

* {{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 }} <!--{{sfn|Burzyk|Gilsdorf|1995|pp=}}-->
* {{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}} <!--{{sfn|Diestel|2008|p=}}-->
* {{Dubinsky The Structure of Nuclear Fréchet Spaces}} <!--{{sfn|Dubinsky|1979|p=}}-->
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!--{{sfn|Grothendieck|1955|p=}}-->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!--{{sfn|Hogbe-Nlend|1977|p=}}-->
* {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}} <!--{{sfn|Hogbe-Nlend|Moscatelli|1981|p=}}-->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!--{{sfn|Husain|Khaleelulla|1978|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Pietsch Nuclear Locally Convex Spaces|edition=2}} <!--{{sfn|Pietsch|1979|p=}}-->
* {{Robertson Topological Vector Spaces}} <!--{{sfn|Robertson|Robertson|1980|p=}}-->
* {{Ryan Introduction to Tensor Products of Banach Spaces|edition=1}} <!--{{sfn|Ryan|2002|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->
* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!--{{sfn|Wong|1979|p=}}-->

==External links==

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

{{Functional analysis}}
{{TopologicalTensorProductsAndNuclearSpaces}}

<!--- Categories --->
[[Category:Functional analysis]]

{{Sourceattribution|Auxiliary normed space}}