nLab inductive tensor product

Idea

In functional analysis, the inductive tensor product is a kind of tensor product suitable for topological vector spaces.

Specifically, the inductive tensor product $E\otimes_i F$ of a pair of topological vector spaces $E$, and $F$ is the target of the universal separately continuous bilinear map $E\times F \to E\otimes_i F$.

Definition

Given a pair of locally convex topological vector spaces (lctvs) $E$ and $F$, the inductive tensor product is the locally convex tvs $E\otimes_i F$ with underlying plain vector space the algebraic tensor product of vector spaces, and equipped with the finest locally convex linear topology such that $E\times F \to E\otimes_i F$ is separately continuous, where $E\times F$ has the product topology. The definition is due to (Grothendieck 1955, Définition 3 in I \S 3.I)

Compare this to the (pre-completed) projective tensor product $\otimes_p$, which is equipped with the finest locally convex linear topology such that the canonical map from $E\times F$ is jointly continuous.

Sometimes one will wish to consider this tensor product in the category of complete lctvs, and so the result is the topological completion $E\overline{\otimes} F$ of $E\otimes_i F$ (eg by adding in the limits of Cauchy nets), which is the analogue of the (complete) projective tensor product $\widehat{\otimes}$.

Properties

Let $lctvs$ denote the category of locally convex topological vector spaces and bounded maps, and $lcvts_{comp}$ the reflective subcategory of complete lctvs.

Theorem

The inductive tensor product functor $\otimes_i\colon lctvs \times lctvs \to lctvs$ separately preserves inductive limits. Moreover $\otimes_i$ equips $lctvs$ with the structure of a symmetric monoidal category, and even a symmetric closed monoidal category, by equipping the space of continuous maps with the topology of pointwise convergence.

Be aware though that the completed inductive tensor product $\overline{\otimes}$ does not make $lctvs_{comp}$ into a monoidal category, as coherent associativity fails.

Proposition

Let $E$ and $F$ be metrizable tvs, and $T\colon E\times F \to G$ a separately continuous bilinear function. If one of the following conditions is true

1. $E$ is a Baire space (for example, completely metrizable)
2. $E$ is a barreled topological vector space (eg is a Frechet space) and $G$ is locally convex.

then $T$ is jointly continuous.

For a proof see (Schaefer-Wolff 1999, corollary to Theorem 5.1)

Notice that there are canonical comparison maps $E\otimes_i F \to E\otimes_p F$ and $E\overline{\otimes} F \to E \widehat{\otimes} F$.

Theorem

Let $E$ and $F$ be metrizable lctvs, and assume one of the conditions in Proposition holds. Then the comparison maps from the inductive tensor products to the projective tensor products are isomorphisms.

Corollary

If $E$ and $F$ are Fréchet spaces, then $E\overline{\otimes} F \stackrel{\sim}{\to} E \widehat{\otimes} F$.

Hence for Banach spaces and Hilbert spaces the inductive tensor product also agrees with the projective tensor product. It is for this reason that one can define locally convex algebra?s with the inductive tensor product, as this reduces the the usual jointly-continuous multiplication for Banach algebras, $C^*$-algebras etc.

Related pages

• spatial tensor product
• tensor product of Banach spaces

References

• Alexandre Grothendieck, Produits tensoriels topologiques et spaces nucléaires, Memoirs AMS 16 (1955)

• Schaeffer, Wolff, Topological Vector Spaces, Graduate Texts in Mathematics 3 (1999) Springer.