Topological tensor product

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