Projective tensor product

In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces $$X$$ and $$Y$$, the projective topology, or π-topology, on $$X \otimes Y$$ is the strongest topology which makes $$X \otimes Y$$ a locally convex topological vector space such that the canonical map $$(x,y) \mapsto x \otimes y$$ (from $$X\times Y$$ to $$X \otimes Y$$) is continuous. When equipped with this topology, $$X \otimes Y$$ is denoted $$X \otimes_\pi Y$$ and called the projective tensor product of $$X$$ and $$Y$$.

Definitions
Let $$X$$ and $$Y$$ be locally convex topological vector spaces. Their projective tensor product $$X \otimes_\pi Y$$ is the unique locally convex topological vector space with underlying vector space $$X \otimes Y$$ having the following universal property:
 * For any locally convex topological vector space $$Z$$, if $$\Phi_Z$$ is the canonical map from the vector space of bilinear maps $$X\times Y \to Z$$ to the vector space of linear maps $$X \otimes Y \to Z$$, then the image of the restriction of $$\Phi_Z$$ to the continuous bilinear maps is the space of continuous linear maps $$X \otimes_\pi Y \to Z$$.

When the topologies of $$X$$ and $$Y$$ are induced by seminorms, the topology of $$X \otimes_\pi Y$$ is induced by seminorms constructed from those on $$X$$ and $$Y$$ as follows. If $$p$$ is a seminorm on $$X$$, and $$q$$ is a seminorm on $$Y$$, define their tensor product $$p \otimes q$$ to be the seminorm on $$X \otimes Y$$ given by $$(p \otimes q)(b) = \inf_{r > 0,\, b \in r W} r$$ for all $$b$$ in $$X \otimes Y$$, where $$W$$ is the balanced convex hull of the set $$\left\{ x \otimes y : p(x) \leq 1, q(y) \leq 1 \right\}$$. The projective topology on $$X \otimes Y$$ is generated by the collection of such tensor products of the seminorms on $$X$$ and $$Y$$. When $$X$$ and $$Y$$ are normed spaces, this definition applied to the norms on $$X$$ and $$Y$$ gives a norm, called the projective norm, on $$X \otimes Y$$ which generates the projective topology.

Properties
Throughout, all spaces are assumed to be locally convex. The symbol $$X \widehat{\otimes}_\pi Y$$ denotes the completion of the projective tensor product of $$X$$ and $$Y$$.
 * If $$X$$ and $$Y$$ are both Hausdorff then so is $$X \otimes_\pi Y$$; if $$X$$ and $$Y$$ are Fréchet spaces then $$X \otimes_\pi Y$$ is barelled.
 * For any two continuous linear operators $$u_1 : X_1 \to Y_1$$ and $$u_2 : X_2 \to Y_2$$, their tensor product (as linear maps) $$u_1 \otimes u_2 : X_1 \otimes_\pi X_2 \to Y_1 \otimes_\pi Y_2$$ is continuous.
 * In general, the projective tensor product does not respect subspaces (e.g. if $$Z$$ is a vector subspace of $$X$$ then the TVS $$Z \otimes_\pi Y$$ has in general a coarser topology than the subspace topology inherited from $$X \otimes_\pi Y$$).
 * If $$E$$ and $$F$$ are complemented subspaces of $$X$$ and $$Y,$$ respectively, then $$E \otimes F$$ is a complemented vector subspace of $$X \otimes_\pi Y$$ and the projective norm on $$E \otimes_\pi F$$ is equivalent to the projective norm on $$X \otimes_\pi Y$$ restricted to the subspace $$E \otimes F$$. Furthermore, if $$X$$ and $$F$$ are complemented by projections of norm 1, then $$E \otimes F$$ is complemented by a projection of norm 1.
 * Let $$E$$ and $$F$$ be vector subspaces of the Banach spaces $$X$$ and $$Y$$, respectively. Then $$E \widehat{\otimes} F$$ is a TVS-subspace of $$X \widehat{\otimes}_\pi Y$$ if and only if every bounded bilinear form on $$E \times F$$ extends to a continuous bilinear form on $$X \times Y$$ with the same norm.

Completion
In general, the space $$X \otimes_\pi Y$$ is not complete, even if both $$X$$ and $$Y$$ are complete (in fact, if $$X$$ and $$Y$$ are both infinite-dimensional Banach spaces then $$X \otimes_\pi Y$$ is necessarily complete). However, $$X \otimes_\pi Y$$ can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by $$X \widehat{\otimes}_\pi Y$$.

The continuous dual space of $$X \widehat{\otimes}_\pi Y$$ is the same as that of $$X \otimes_\pi Y$$, namely, the space of continuous bilinear forms $$B(X, Y)$$.

Grothendieck's representation of elements in the completion
In a Hausdorff locally convex space $$X,$$ a sequence $$\left(x_i\right)_{i=1}^{\infty}$$ in $$X$$ is absolutely convergent if $$\sum_{i=1}^{\infty} p \left(x_i\right) < \infty$$ for every continuous seminorm $$p$$ on $$X.$$ We write $$x = \sum_{i=1}^{\infty} x_i$$ if the sequence of partial sums $$\left(\sum_{i=1}^n x_i\right)_{n=1}^{\infty}$$ converges to $$x$$ in $$X.$$

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.

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The next theorem shows that it is possible to make the representation of $$z$$ independent of the sequences $$\left(x_i\right)_{i=1}^{\infty}$$ and $$\left(y_i\right)_{i=1}^{\infty}.$$

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Topology of bi-bounded convergence
Let $$\mathfrak{B}_X$$ and $$\mathfrak{B}_Y$$ denote the families of all bounded subsets of $$X$$ and $$Y,$$ respectively. Since the continuous dual space of $$X \widehat{\otimes}_\pi Y$$ is the space of continuous bilinear forms $$B(X, Y),$$ we can place on $$B(X, Y)$$ the topology of uniform convergence on sets in $$\mathfrak{B}_X \times \mathfrak{B}_Y,$$ which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on $$B(X, Y)$$, and in, Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset $$B \subseteq X \widehat{\otimes} Y,$$ do there exist bounded subsets $$B_1 \subseteq X$$ and $$B_2 \subseteq Y$$ such that $$B$$ is a subset of the closed convex hull of $$B_1 \otimes B_2 := \{ b_1 \otimes b_2 : b_1 \in B_1, b_2 \in B_2 \}$$?

Grothendieck proved that these topologies are equal when $$X$$ and $$Y$$ are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck). They are also equal when both spaces are Fréchet with one of them being nuclear.

Strong dual and bidual
Let $$X$$ be a locally convex topological vector space and let $$X^{\prime}$$ be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

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Examples

 * For $$(X, \mathcal{A}, \mu)$$ a measure space, let $$L^1$$ be the real Lebesgue space $$L^1(\mu)$$; let $$E$$ be a real Banach space. Let $$L^1_E$$ be the completion of the space of simple functions $$X\to E$$, modulo the subspace of functions $$X\to E$$ whose pointwise norms, considered as functions $$X\to\Reals$$, have integral $$0$$ with respect to $$\mu$$. Then $$L^1_E$$ is isometrically isomorphic to $$L^1 \widehat{\otimes}_\pi E$$.