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A [[Locally convex space|locally convex space]] having a universality property with respect to bilinear operators on $E_1 \times E_2$ and satisfying a continuity condition. More precisely, let $\mathcal{K}$ be a certain class of locally convex spaces and for each $F \in \mathcal{K}$ let there be given a subset $T(F)$ of the set of separately-continuous bilinear operators from $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \tilde\otimes E_2 \in \mathcal{K}$ together with the operator $B \in T(E_1 \tilde\otimes E_2)$ having the following property: For any $S \in T(F)$, $F \in \mathcal{K}$, there exists a unique continuous linear operator $R:E_1 \tilde\otimes E_2 \to F$ such that $R \circ B = S$. Thus, if one speaks of the functor $T: \mathcal{K} \to \mathrm{Sets}$, then $E_1 \tilde\otimes E_2$ is defined as the representing object of this functor.
 
A [[Locally convex space|locally convex space]] having a universality property with respect to bilinear operators on $E_1 \times E_2$ and satisfying a continuity condition. More precisely, let $\mathcal{K}$ be a certain class of locally convex spaces and for each $F \in \mathcal{K}$ let there be given a subset $T(F)$ of the set of separately-continuous bilinear operators from $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \tilde\otimes E_2 \in \mathcal{K}$ together with the operator $B \in T(E_1 \tilde\otimes E_2)$ having the following property: For any $S \in T(F)$, $F \in \mathcal{K}$, there exists a unique continuous linear operator $R:E_1 \tilde\otimes E_2 \to F$ such that $R \circ B = S$. Thus, if one speaks of the functor $T: \mathcal{K} \to \mathrm{Sets}$, then $E_1 \tilde\otimes E_2$ is defined as the representing object of this functor.
  
In all known examples $\mathcal{K}$ contains the field of complex numbers $\mathbb{C}$, and $T(\mathbb{C})$ contains all bilinear functionals of the form $f \circ g$, $f \in E_1^\ast$, $g \in E_2^\ast$, mapping $(x,y)$ to $f(x)g(y)$. If in this case the topological tensor product exists, then there is a dense subspace in $E_1 \tilde\otimes E_2$ that can be identified with the algebraic [[Tensor product|tensor product]] $E_1 \otimes E_2$; moreover, $B(x,y)=x \otimes y$.
+
In all known examples $\mathcal{K}$ contains the field of complex numbers $\mathbb{C}$, and $T(\mathbb{C})$ contains all bilinear functionals of the form $f \circ g$, $f \in E_1^\ast$, $g \in E_2^\ast$, mapping $(x,y)$ to $f(x)g(y)$. If in this case the topological tensor product exists, then there is a dense subspace in $E_1 \tilde\otimes E_2$ that can be identified with the algebraic [[Tensor product|tensor product]] $E_1 {\otimes} E_2$; moreover, $B(x,y)=x {\otimes} y$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316031.png" /> consists of all separately (respectively, jointly) continuous bilinear operators, then the topological tensor product is called inductive (respectively, projective). The most important is the projective topological tensor product. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316032.png" /> be a defining family of semi-norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316034.png" />; denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316035.png" /> the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316036.png" /> defined by the family of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316037.png" />:
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If $T(\mathcal{K})$ consists of all separately (respectively, jointly) continuous bilinear operators, then the topological tensor product is called inductive (respectively, projective). The most important is the projective topological tensor product. Let $\{p_i\}$ be a defining family of semi-norms in $E_i$, $i=1,2$; denote by $\pi$ the topology on $E_1 {\otimes} E_2$ defined by the family of semi-norms $\{p_1 \hat \otimes p_2\}$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316038.png" /></td> </tr></table>
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$$p_1 \hat \otimes p_2(u) = \inf \left\{ \sum_{k=1}^n p_1(x_k) p_2(y_k) : \sum_{k=1}^n x_k{\otimes} y_k = u \right\}.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316039.png" /> is the class of all, respectively all complete, locally convex spaces, then the projective topological tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316041.png" /> exists and its locally convex space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316042.png" /> with the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316043.png" />, respectively its [[Completion|completion]]. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316044.png" /> are Banach spaces with norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316047.png" /> is a norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316048.png" />; the completion with respect to it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316049.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316050.png" /> the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316051.png" /> have the representation
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If $\mathcal{K}$ is the class of all, respectively all complete, locally convex spaces, then the projective topological tensor product of $E_1$ and $E_2$ exists and its locally convex space is $E_1 {\otimes} E_2$ with the topology $\pi$, respectively its [[Completion|completion]]. If the $E_i$ are Banach spaces with norms $p_i$, $i=1,2$, then $p_1 \hat \otimes p_2$ is a norm on $E_1 {\otimes} E_2$; the completion with respect to it is denoted by $E_1 \hat \otimes E_2$. For each $\epsilon>0$ the elements of $E_1 \hat \otimes E_2$ have the representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316052.png" /></td> </tr></table>
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$$ u= \sum_{k=1}^\infty x_k{\otimes} y_k $$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316053.png" /></td> </tr></table>
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$$ \sum_{k=1}^\infty p_1(x_k) p_2(y_k) \leq p_1 \hat \otimes p_2 (u) + \epsilon. $$
  
If one endows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316054.png" /> with a topology weaker than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316055.png" /> by using the family of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316056.png" />,
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If one endows $E_1 {\otimes} E_2$ with a topology weaker than $\pi$ by using the family of semi-norms $p_1 \tilde\otimes p_2$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316057.png" /></td> </tr></table>
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$$p_1 \tilde \otimes p_2(u) = \sup_{f,g \in V \times W} \left\lvert (f {\otimes} g) (u) \right\rvert,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316059.png" /> are the polar sets of the unit spheres with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316061.png" />, then there arises a topological tensor product, sometimes called injective. The locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316062.png" /> with the property that for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316063.png" /> both topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316064.png" /> coincide, form the important class of nuclear spaces (cf. [[Nuclear space|Nuclear space]]).
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where $V$ and $W$ are the polar sets of the unit spheres with respect to $p_1$ and $p_2$, then there arises a topological tensor product, sometimes called injective. The locally convex spaces $E_1$ with the property that for an arbitrary $E_2$ both topologies in $E_1 {\otimes} E_2$ coincide, form the important class of nuclear spaces (cf. [[Nuclear space|Nuclear space]]).
  
The projective topological tensor product is associated with the approximation property: A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316065.png" /> has the approximation property if for each pre-compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316066.png" /> and neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316067.png" /> there exists a continuous operator of finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316068.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316069.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316070.png" />. All nuclear spaces have the approximation property. A Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316071.png" /> has the approximation property if and only if for an arbitrary Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316072.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316073.png" />, unambiguously defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316074.png" />, has trivial kernel. A separable Banach space without the approximation property has been constructed [[#References|[3]]]. This space also gives an example of a Banach space without a Schauder basis, since the Banach spaces with a Schauder basis have the approximation property (thus S. Banach's so-called  "Banach basis problembasis problem" has been negatively solved).
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The projective topological tensor product is associated with the approximation property: A locally convex space $E_1$ has the approximation property if for each pre-compact set $K \subset E_1$ and neighbourhood of zero $U$ there exists a continuous operator of finite rank $\phi:E_1 \to E_1$ such that for all $x \in K$ one has $x-\phi(x) \in U$. All nuclear spaces have the approximation property. A Banach space $E_1$ has the approximation property if and only if for an arbitrary Banach space $E_2$ the operator $\tau: E_1 \hat\otimes E_2 \to (E_1^\ast{\otimes}E_2^\ast)^\ast$, unambiguously defined by the equation $\tau(x {\otimes} y) (f {\otimes} g)=f(x)g(y)$, has trivial kernel. A separable Banach space without the approximation property has been constructed [[#References|[3]]]. This space also gives an example of a Banach space without a Schauder basis, since the Banach spaces with a Schauder basis have the approximation property (thus S. Banach's so-called  "Banach basis problem" has been negatively solved).
  
 
====References====
 
====References====

Latest revision as of 23:16, 10 April 2019

of two locally convex spaces $E_1$ and $E_2$

A locally convex space having a universality property with respect to bilinear operators on $E_1 \times E_2$ and satisfying a continuity condition. More precisely, let $\mathcal{K}$ be a certain class of locally convex spaces and for each $F \in \mathcal{K}$ let there be given a subset $T(F)$ of the set of separately-continuous bilinear operators from $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \tilde\otimes E_2 \in \mathcal{K}$ together with the operator $B \in T(E_1 \tilde\otimes E_2)$ having the following property: For any $S \in T(F)$, $F \in \mathcal{K}$, there exists a unique continuous linear operator $R:E_1 \tilde\otimes E_2 \to F$ such that $R \circ B = S$. Thus, if one speaks of the functor $T: \mathcal{K} \to \mathrm{Sets}$, then $E_1 \tilde\otimes E_2$ is defined as the representing object of this functor.

In all known examples $\mathcal{K}$ contains the field of complex numbers $\mathbb{C}$, and $T(\mathbb{C})$ contains all bilinear functionals of the form $f \circ g$, $f \in E_1^\ast$, $g \in E_2^\ast$, mapping $(x,y)$ to $f(x)g(y)$. If in this case the topological tensor product exists, then there is a dense subspace in $E_1 \tilde\otimes E_2$ that can be identified with the algebraic tensor product $E_1 {\otimes} E_2$; moreover, $B(x,y)=x {\otimes} y$.

If $T(\mathcal{K})$ consists of all separately (respectively, jointly) continuous bilinear operators, then the topological tensor product is called inductive (respectively, projective). The most important is the projective topological tensor product. Let $\{p_i\}$ be a defining family of semi-norms in $E_i$, $i=1,2$; denote by $\pi$ the topology on $E_1 {\otimes} E_2$ defined by the family of semi-norms $\{p_1 \hat \otimes p_2\}$:

$$p_1 \hat \otimes p_2(u) = \inf \left\{ \sum_{k=1}^n p_1(x_k) p_2(y_k) : \sum_{k=1}^n x_k{\otimes} y_k = u \right\}.$$

If $\mathcal{K}$ is the class of all, respectively all complete, locally convex spaces, then the projective topological tensor product of $E_1$ and $E_2$ exists and its locally convex space is $E_1 {\otimes} E_2$ with the topology $\pi$, respectively its completion. If the $E_i$ are Banach spaces with norms $p_i$, $i=1,2$, then $p_1 \hat \otimes p_2$ is a norm on $E_1 {\otimes} E_2$; the completion with respect to it is denoted by $E_1 \hat \otimes E_2$. For each $\epsilon>0$ the elements of $E_1 \hat \otimes E_2$ have the representation

$$ u= \sum_{k=1}^\infty x_k{\otimes} y_k $$

where

$$ \sum_{k=1}^\infty p_1(x_k) p_2(y_k) \leq p_1 \hat \otimes p_2 (u) + \epsilon. $$

If one endows $E_1 {\otimes} E_2$ with a topology weaker than $\pi$ by using the family of semi-norms $p_1 \tilde\otimes p_2$,

$$p_1 \tilde \otimes p_2(u) = \sup_{f,g \in V \times W} \left\lvert (f {\otimes} g) (u) \right\rvert,$$

where $V$ and $W$ are the polar sets of the unit spheres with respect to $p_1$ and $p_2$, then there arises a topological tensor product, sometimes called injective. The locally convex spaces $E_1$ with the property that for an arbitrary $E_2$ both topologies in $E_1 {\otimes} E_2$ coincide, form the important class of nuclear spaces (cf. Nuclear space).

The projective topological tensor product is associated with the approximation property: A locally convex space $E_1$ has the approximation property if for each pre-compact set $K \subset E_1$ and neighbourhood of zero $U$ there exists a continuous operator of finite rank $\phi:E_1 \to E_1$ such that for all $x \in K$ one has $x-\phi(x) \in U$. All nuclear spaces have the approximation property. A Banach space $E_1$ has the approximation property if and only if for an arbitrary Banach space $E_2$ the operator $\tau: E_1 \hat\otimes E_2 \to (E_1^\ast{\otimes}E_2^\ast)^\ast$, unambiguously defined by the equation $\tau(x {\otimes} y) (f {\otimes} g)=f(x)g(y)$, has trivial kernel. A separable Banach space without the approximation property has been constructed [3]. This space also gives an example of a Banach space without a Schauder basis, since the Banach spaces with a Schauder basis have the approximation property (thus S. Banach's so-called "Banach basis problem" has been negatively solved).

References

[1] A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955)
[2] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
[3] P. Enflo, "A counterexample to the approximation problem in Banach spaces" Acta Math. , 130 (1973) pp. 309–317


Comments

References

[a1] A. Pietsch, "Nukleare lokalkonvexe Räume" , Akademie Verlag (1965)
[a2] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977)
[a3] F. Trèves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967) pp. 198
[a4] G. Pisier, "Factorisation of linear operators and geometry of Banach spaces" , Amer. Math. Soc. (1986)
How to Cite This Entry:
Topological tensor product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_tensor_product&oldid=43772
This article was adapted from an original article by A.Ya. Khelemskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article