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:^[1]

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 $(p\otimes q)(b)=\inf _{r>0,\,b\in rW}r$ 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 $Y$ .^[2]^[1] 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.^[3]

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 $X\otimes _{\pi }Y$ ;^[3] if $X$ and $Y$ are Fréchet spaces then $X\otimes _{\pi }Y$ is barelled.^[4]
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.^[5]
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 $X\otimes _{\pi }Y$ ).^[6]
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.^[6]
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.^[7]

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 not complete^[8]). 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 $B(X,Y)$ .^[9]

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 $X.$ ^[10] 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 $X.$ ^[10]

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.^[11]

Theorem — Let $X$ and $Y$ be metrizable locally convex TVSs and let $z\in X{\widehat {\otimes }}_{\pi }Y.$ Then $z$ is the sum of an absolutely convergent series $z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}$ where $\sum _{i=1}^{\infty }|\lambda _{i}|<\infty ,$ and $\left(x_{i}\right)_{i=1}^{\infty }$ and $\left(y_{i}\right)_{i=1}^{\infty }$ are null sequences in $X$ and $Y,$ respectively.

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 $\left(y_{i}\right)_{i=1}^{\infty }.$

Theorem^[12] — Let $X$ and $Y$ be Fréchet spaces and let $U$ (resp. $V$ ) be a balanced open neighborhood of the origin in $X$ (resp. in $Y$ ). Let $K_{0}$ be a compact subset of the convex balanced hull of $U\otimes V:=\{u\otimes v:u\in U,v\in V\}.$ There exists a compact subset $K_{1}$ of the unit ball in $\ell ^{1}$ and sequences $\left(x_{i}\right)_{i=1}^{\infty }$ and $\left(y_{i}\right)_{i=1}^{\infty }$ contained in $U$ and $V,$ respectively, converging to the origin such that for every $z\in K_{0}$ there exists some $\left(\lambda _{i}\right)_{i=1}^{\infty }\in K_{1}$ such that $z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}.$

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 (Grothendieck 1955), 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^[13]). They are also equal when both spaces are Fréchet with one of them being nuclear.^[9]

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:

Theorem^[14] (Grothendieck) — Let $N$ and $Y$ be locally convex topological vector spaces with $N$ nuclear. Assume that both $N$ and $Y$ are Fréchet spaces, or else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted $b$ :

The strong dual of $N{\widehat {\otimes }}_{\pi }Y$ can be identified with $N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }$ ;
The bidual of $N{\widehat {\otimes }}_{\pi }Y$ can be identified with $N{\widehat {\otimes }}_{\pi }Y^{\prime \prime }$ ;
If $Y$ is reflexive then $N{\widehat {\otimes }}_{\pi }Y$ (and hence $N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }$ ) is a reflexive space;
Every separately continuous bilinear form on $N_{b}^{\prime }\times Y_{b}^{\prime }$ is continuous;
Let $L\left(X_{b}^{\prime },Y\right)$ be the space of bounded linear maps from $X_{b}^{\prime }$ to $Y$ . Then, its strong dual can be identified with $N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime },$ so in particular if $Y$ is reflexive then so is $L_{b}\left(X_{b}^{\prime },Y\right).$

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 $L_{E}^{1}$ 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 $X\to \mathbb {R}$ , have integral $0$ with respect to $\mu$ . Then $L_{E}^{1}$ is isometrically isomorphic to $L^{1}{\widehat {\otimes }}_{\pi }E$ .^[15]

Citations

^ ^a ^b Trèves 2006, p. 438.
^ Trèves 2006, p. 435.
^ ^a ^b Trèves 2006, p. 437.
^ Trèves 2006, p. 445.
^ Trèves 2006, p. 439.
^ ^a ^b Ryan 2002, p. 18.
^ Ryan 2002, p. 24.
^ Ryan 2002, p. 43.
^ ^a ^b Schaefer & Wolff 1999, p. 173.
^ ^a ^b Schaefer & Wolff 1999, p. 120.
^ Schaefer & Wolff 1999, p. 94.
^ Trèves 2006, pp. 459–460.
^ Schaefer & Wolff 1999, p. 154.
^ Schaefer & Wolff 1999, pp. 175–176.
^ Schaefer & Wolff 1999, p. 95.

References

Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

External links

Nuclear space at ncatlab

[FOOTNOTETrèves2006438-1] Trèves 2006, p. 438.

[FOOTNOTETrèves2006435-2] Trèves 2006, p. 435.

[FOOTNOTETrèves2006437-3] Trèves 2006, p. 437.

[FOOTNOTETrèves2006445-4] Trèves 2006, p. 445.

[FOOTNOTETrèves2006439-5] Trèves 2006, p. 439.

[FOOTNOTERyan200218-6] Ryan 2002, p. 18.

[FOOTNOTERyan200224-7] Ryan 2002, p. 24.

[FOOTNOTERyan200243-8] Ryan 2002, p. 43.

[FOOTNOTESchaeferWolff1999173-9] Schaefer & Wolff 1999, p. 173.

[FOOTNOTESchaeferWolff1999120-10] Schaefer & Wolff 1999, p. 120.

[FOOTNOTESchaeferWolff199994-11] Schaefer & Wolff 1999, p. 94.

[FOOTNOTETrèves2006459–460-12] Trèves 2006, pp. 459–460.

[FOOTNOTESchaeferWolff1999154-13] Schaefer & Wolff 1999, p. 154.

[FOOTNOTESchaeferWolff1999175–176-14] Schaefer & Wolff 1999, pp. 175–176.

[FOOTNOTESchaeferWolff199995-15] Schaefer & Wolff 1999, p. 95.

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Topological tensor products and nuclear spaces
Basic concepts	Auxiliary normed spaces Nuclear space Tensor product Topological tensor product of Hilbert spaces
Topologies	Inductive tensor product Injective tensor product Projective tensor product
Operators/Maps	Fredholm determinant Fredholm kernel Hilbert–Schmidt operator Hypocontinuity Integral Nuclear between Banach spaces Trace class
Theorems	Grothendieck trace theorem Schwartz kernel theorem

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