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 [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], the projective topology, or π-topology, on [math]\displaystyle{ X \otimes Y }[/math] is the strongest topology which makes [math]\displaystyle{ X \otimes Y }[/math] a locally convex topological vector space such that the canonical map [math]\displaystyle{ (x,y) \mapsto x \otimes y }[/math] (from [math]\displaystyle{ X\times Y }[/math] to [math]\displaystyle{ X \otimes Y }[/math]) is continuous. When equipped with this topology, [math]\displaystyle{ X \otimes Y }[/math] is denoted [math]\displaystyle{ X \otimes_\pi Y }[/math] and called the projective tensor product of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math].

Definitions

Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be locally convex topological vector spaces. Their projective tensor product [math]\displaystyle{ X \otimes_\pi Y }[/math] is the unique locally convex topological vector space with underlying vector space [math]\displaystyle{ X \otimes Y }[/math] having the following universal property:^[1]

For any locally convex topological vector space [math]\displaystyle{ Z }[/math], if [math]\displaystyle{ \Phi_Z }[/math] is the canonical map from the vector space of bilinear maps [math]\displaystyle{ X\times Y \to Z }[/math] to the vector space of linear maps [math]\displaystyle{ X \otimes Y \to Z }[/math]; then the image of the restriction of [math]\displaystyle{ \Phi_Z }[/math] to the continuous bilinear maps is the space of continuous linear maps [math]\displaystyle{ X \otimes_\pi Y \to Z }[/math].

When the topologies of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are induced by seminorms, the topology of [math]\displaystyle{ X \otimes_\pi Y }[/math] is induced by seminorms constructed from those on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] as follows. If [math]\displaystyle{ p }[/math] is a seminorm on [math]\displaystyle{ X }[/math], and [math]\displaystyle{ q }[/math] is a seminorm on [math]\displaystyle{ Y }[/math], define their tensor product [math]\displaystyle{ p \otimes q }[/math] to be the seminorm on [math]\displaystyle{ X \otimes Y }[/math] given by [math]\displaystyle{ (p \otimes q)(b) = \inf_{r \gt 0,\, b \in r W} r }[/math] for all [math]\displaystyle{ b }[/math] in [math]\displaystyle{ X \otimes Y }[/math], where [math]\displaystyle{ W }[/math] is the balanced convex hull of the set [math]\displaystyle{ \left\{ x \otimes y : p(x) \leq 1, q(y) \leq 1 \right\} }[/math]. The projective topology on [math]\displaystyle{ X \otimes Y }[/math] is generated by the collection of such tensor products of the seminorms on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math].^[2]^[1] When [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are normed spaces, this definition applied to the norms on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] gives a norm, called the projective norm, on [math]\displaystyle{ X \otimes Y }[/math] which generates the projective topology.^[3]

Properties

Throughout, all spaces are assumed to be locally convex. The symbol [math]\displaystyle{ X \widehat{\otimes}_\pi Y }[/math] denotes the completion of the projective tensor product of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math].

If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are both Hausdorff then so is [math]\displaystyle{ X \otimes_\pi Y }[/math];^[3] if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are Fréchet spaces then [math]\displaystyle{ X \otimes_\pi Y }[/math] is barelled.^[4]
For any two continuous linear operators [math]\displaystyle{ u_1 : X_1 \to Y_1 }[/math] and [math]\displaystyle{ u_2 : X_2 \to Y_2 }[/math], their tensor product (as linear maps) [math]\displaystyle{ u_1 \otimes u_2 : X_1 \otimes_\pi X_2 \to Y_1 \otimes_\pi Y_2 }[/math] is continuous.^[5]
In general, the projective tensor product does not respect subspaces (e.g. if [math]\displaystyle{ Z }[/math] is a vector subspace of [math]\displaystyle{ X }[/math] then the TVS [math]\displaystyle{ Z \otimes_\pi Y }[/math] has in general a coarser topology than the subspace topology inherited from [math]\displaystyle{ X \otimes_\pi Y }[/math]).^[6]
If [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math] are complemented subspaces of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y, }[/math] respectively, then [math]\displaystyle{ E \otimes F }[/math] is a complemented vector subspace of [math]\displaystyle{ X \otimes_\pi Y }[/math] and the projective norm on [math]\displaystyle{ E \otimes_\pi F }[/math] is equivalent to the projective norm on [math]\displaystyle{ X \otimes_\pi Y }[/math] restricted to the subspace [math]\displaystyle{ E \otimes F }[/math]. Furthermore, if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ F }[/math] are complemented by projections of norm 1, then [math]\displaystyle{ E \otimes F }[/math] is complemented by a projection of norm 1.^[6]
Let [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math] be vector subspaces of the Banach spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], respectively. Then [math]\displaystyle{ E \widehat{\otimes} F }[/math] is a TVS-subspace of [math]\displaystyle{ X \widehat{\otimes}_\pi Y }[/math] if and only if every bounded bilinear form on [math]\displaystyle{ E \times F }[/math] extends to a continuous bilinear form on [math]\displaystyle{ X \times Y }[/math] with the same norm.^[7]

Completion

In general, the space [math]\displaystyle{ X \otimes_\pi Y }[/math] is not complete, even if both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are complete (in fact, if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are both infinite-dimensional Banach spaces then [math]\displaystyle{ X \otimes_\pi Y }[/math] is necessarily not complete^[8]). However, [math]\displaystyle{ X \otimes_\pi Y }[/math] can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by [math]\displaystyle{ X \widehat{\otimes}_\pi Y }[/math].

The continuous dual space of [math]\displaystyle{ X \widehat{\otimes}_\pi Y }[/math] is the same as that of [math]\displaystyle{ X \otimes_\pi Y }[/math], namely, the space of continuous bilinear forms [math]\displaystyle{ B(X, Y) }[/math].^[9]

Grothendieck's representation of elements in the completion

In a Hausdorff locally convex space [math]\displaystyle{ X, }[/math] a sequence [math]\displaystyle{ \left(x_i\right)_{i=1}^{\infty} }[/math] in [math]\displaystyle{ X }[/math] is absolutely convergent if [math]\displaystyle{ \sum_{i=1}^{\infty} p \left(x_i\right) \lt \infty }[/math] for every continuous seminorm [math]\displaystyle{ p }[/math] on [math]\displaystyle{ X. }[/math]^[10] We write [math]\displaystyle{ x = \sum_{i=1}^{\infty} x_i }[/math] if the sequence of partial sums [math]\displaystyle{ \left(\sum_{i=1}^n x_i\right)_{n=1}^{\infty} }[/math] converges to [math]\displaystyle{ x }[/math] in [math]\displaystyle{ X. }[/math]^[10]

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

Theorem — Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be metrizable locally convex TVSs and let [math]\displaystyle{ z \in X \widehat{\otimes}_\pi Y. }[/math] Then [math]\displaystyle{ z }[/math] is the sum of an absolutely convergent series [math]\displaystyle{ z = \sum_{i=1}^{\infty} \lambda_i x_i \otimes y_i }[/math] where [math]\displaystyle{ \sum_{i=1}^{\infty}|\lambda_i|\lt \infty, }[/math] and [math]\displaystyle{ \left(x_i\right)_{i=1}^{\infty} }[/math] and [math]\displaystyle{ \left(y_i\right)_{i=1}^{\infty} }[/math] are null sequences in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y, }[/math] respectively.

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

Theorem^[12] — Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be Fréchet spaces and let [math]\displaystyle{ U }[/math] (resp. [math]\displaystyle{ V }[/math]) be a balanced open neighborhood of the origin in [math]\displaystyle{ X }[/math] (resp. in [math]\displaystyle{ Y }[/math]). Let [math]\displaystyle{ K_0 }[/math] be a compact subset of the convex balanced hull of [math]\displaystyle{ U \otimes V := \{ u \otimes v : u \in U, v \in V \}. }[/math] There exists a compact subset [math]\displaystyle{ K_1 }[/math] of the unit ball in [math]\displaystyle{ \ell^1 }[/math] and sequences [math]\displaystyle{ \left(x_i\right)_{i=1}^{\infty} }[/math] and [math]\displaystyle{ \left(y_i\right)_{i=1}^{\infty} }[/math] contained in [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V, }[/math] respectively, converging to the origin such that for every [math]\displaystyle{ z \in K_0 }[/math] there exists some [math]\displaystyle{ \left(\lambda_i\right)_{i=1}^{\infty} \in K_1 }[/math] such that [math]\displaystyle{ z = \sum_{i=1}^{\infty} \lambda_i x_i \otimes y_i. }[/math]

Topology of bi-bounded convergence

Let [math]\displaystyle{ \mathfrak{B}_X }[/math] and [math]\displaystyle{ \mathfrak{B}_Y }[/math] denote the families of all bounded subsets of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y, }[/math] respectively. Since the continuous dual space of [math]\displaystyle{ X \widehat{\otimes}_\pi Y }[/math] is the space of continuous bilinear forms [math]\displaystyle{ B(X, Y), }[/math] we can place on [math]\displaystyle{ B(X, Y) }[/math] the topology of uniform convergence on sets in [math]\displaystyle{ \mathfrak{B}_X \times \mathfrak{B}_Y, }[/math] which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on [math]\displaystyle{ B(X, Y) }[/math], 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 [math]\displaystyle{ B \subseteq X \widehat{\otimes} Y, }[/math] do there exist bounded subsets [math]\displaystyle{ B_1 \subseteq X }[/math] and [math]\displaystyle{ B_2 \subseteq Y }[/math] such that [math]\displaystyle{ B }[/math] is a subset of the closed convex hull of [math]\displaystyle{ B_1 \otimes B_2 := \{ b_1 \otimes b_2 : b_1 \in B_1, b_2 \in B_2 \} }[/math]?

Grothendieck proved that these topologies are equal when [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] 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 [math]\displaystyle{ X }[/math] be a locally convex topological vector space and let [math]\displaystyle{ X^{\prime} }[/math] be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

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

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

Examples

For [math]\displaystyle{ (X, \mathcal{A}, \mu) }[/math] a measure space, let [math]\displaystyle{ L^1 }[/math] be the real Lebesgue space [math]\displaystyle{ L^1(\mu) }[/math]; let [math]\displaystyle{ E }[/math] be a real Banach space. Let [math]\displaystyle{ L^1_E }[/math] be the completion of the space of simple functions [math]\displaystyle{ X\to E }[/math], modulo the subspace of functions [math]\displaystyle{ X\to E }[/math] whose pointwise norms, considered as functions [math]\displaystyle{ X\to\Reals }[/math], have integral [math]\displaystyle{ 0 }[/math] with respect to [math]\displaystyle{ \mu }[/math]. Then [math]\displaystyle{ L^1_E }[/math] is isometrically isomorphic to [math]\displaystyle{ L^1 \widehat{\otimes}_\pi E }[/math].^[15]

Citations

↑ ^1.0 ^1.1 Trèves 2006, p. 438.
↑ Trèves 2006, p. 435.
↑ ^3.0 ^3.1 Trèves 2006, p. 437.
↑ Trèves 2006, p. 445.
↑ Trèves 2006, p. 439.
↑ ^6.0 ^6.1 Ryan 2002, p. 18.
↑ Ryan 2002, p. 24.
↑ Ryan 2002, p. 43.
↑ ^9.0 ^9.1 Schaefer & Wolff 1999, p. 173.
↑ ^10.0 ^10.1 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (August 6, 2006). 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

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Original source: https://en.wikipedia.org/wiki/Projective tensor product. Read more

[FOOTNOTETrèves2006438-1] 1.0 ^1.1 Trèves 2006, p. 438.

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

[FOOTNOTETrèves2006437-3] 3.0 ^3.1 Trèves 2006, p. 437.

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

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

[FOOTNOTERyan200218-6] 6.0 ^6.1 Ryan 2002, p. 18.

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

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

[FOOTNOTESchaeferWolff1999173-9] 9.0 ^9.1 Schaefer & Wolff 1999, p. 173.

[FOOTNOTESchaeferWolff1999120-10] 10.0 ^10.1 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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Projective tensor product

Definitions

Properties

Completion

Grothendieck's representation of elements in the completion

Topology of bi-bounded convergence

Strong dual and bidual

Examples

See also

Citations

References

Further reading

External links

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