nLab
tensor product of Banach spaces

Idea

There are various norms that may be placed on the tensor product of the underlying vector spaces of two Banach spaces; the result is not usually complete, but of course we may take its completion. One of these, the projective tensor product, makes Ban (the category of Banach spaces and short linear maps) into a closed monoidal category, but any of them makes $\mathrm{Ban}$ into a symmetric monoidal category. If we start with Hilbert spaces, then a different choice of norm is needed to make the result into a Hilbert space; then Hilb also becomes a closed symmetric monoidal category.

Definitions

Let $V$ and $W$ be Banach spaces, and let $V\otimes W$ be their tensor product as vector spaces. To define a tensor product of $V$ and $W$ as Banach spaces, we will place a norm on $V\otimes W$, making a normed vector space; the only difference in the following definitions is which norm to use. Then we take the completion $V{\displaystyle \hat{\otimes }}W$, which is a Banach space.

Definition (projective tensor product)

Every element of $V\otimes W$ may be written (in many different ways) as a formal linear combination of formal tensor products of elements of $V$ and $W$ (suppressing the symbol $\otimes$):

$$\sum _i{\alpha }_i{v}_i{w}_i.$$\sum_i \alpha_i v_i w_i .

Let the projective cross norm $\parallel x{\parallel }_{\pi }$ of an element $x$ of $V\otimes W$ be

$$\parallel x{\parallel }_{\pi }≔\inf \{\sum _i\mid {\alpha }_i\mid \parallel v_i{\parallel }_V\parallel w_i{\parallel }_W\mid x=\sum _i{\alpha }_i{v}_i{w}_i\}.$${\|x\|_\pi} \coloneqq \inf \{ \sum_i {|\alpha_i|} {\|v_i\|_V} {\|w_i\|_W} \;|\; x = \sum_i \alpha_i v_i w_i \} .

Then the projective tensor product $V{\displaystyle {\hat{\otimes }}_{\pi }}W$ of $V$ and $W$ is the completion of $V\otimes W$ under the projective cross norm.

Definition (injective tensor product)

If $\lambda$ and $\mu$ are linear functionals on $V$ and $W$ (respectively), then $\lambda \otimes \mu$ is a linear functional on $V\otimes W$. Let the injective cross norm $\parallel x{\parallel }_{ϵ}$ of an element $x$ of $V\otimes W$ be

$$\parallel x{\parallel }_{ϵ}≔\sup \{\mid (\lambda \otimes \mu )x\mid \mid \parallel \lambda {\parallel }_{V^{*}},\parallel \mu {\parallel }_{W^{*}}\le 1\}.$${\|x\|_\epsilon} \coloneqq \sup \{ {|(\lambda \otimes \mu)x|} \;|\; {\|\lambda\|_{V^*}}, {\|\mu\|_{W^*}} \leq 1 \} .

Then the injective tensor product $V{\displaystyle {\hat{\otimes }}_{ϵ}}W$ of $V$ and $W$ is the completion of $V\otimes W$ under the injective cross norm.

Definition (tensor product of Hilbert spaces)

If $V$ and $W$ are Hilbert spaces, then their norms determine and are determined by their inner products, so let us discuss inner products. The elements of $V\otimes W$ are generated by elements of the form $vw$, so set

$$⟨v_1{w}_1,v_2{w}_2⟩≔⟨v_1,v_2⟩⟨w_1,w_2⟩$$\langle{v_1 w_1, v_2 w_2}\rangle \coloneqq \langle{v_1, v_2}\rangle \langle{w_1, w_2}\rangle

and extend by linearity. We write the norm of an element $x$ of the inner product space $V\otimes W$ as $\parallel x{\parallel }_{\sigma }$. Then the tensor product $V{\displaystyle {\hat{\otimes }}_{\sigma }}W$ of the Hilbert spaces $V$ and $W$ is the completion of $V\otimes W$ under this norm (or inner product).

Cross norms

Besides the specific norms defined above, we can define axioms of a reasonable norm on $V\otimes W$.

Definition (cross norm)

A cross norm on $V$ and $W$ is any norm $\chi$ on $V\otimes W$ such that:

• $\parallel v\otimes w{\parallel }_{\chi }=\parallel v{\parallel }_V\parallel w{\parallel }_W$ for any elements $v$ and $w$ of $V$ and $W$ (respectively);
• $\parallel \lambda \otimes \mu {\parallel }_{{\chi }^{*}}=\parallel \lambda {\parallel }_{V^{*}}\parallel \mu {\parallel }_{W^{*}}$ for any bounded linear functionals $\lambda$ and $\mu$ on $V$ and $W$ (respectively).

Definition (uniform cross norm)

A uniform cross norm is an operation that takes two Banach spaces and returns a norm on their algebraic tensor product, naturally in the two spaces. Equivalently, it's a functor $\chi :\mathrm{Ban}\times \mathrm{Ban}\to \mathrm{NVect}$ that makes the following diagram commute (or fills it with a natural isomorphism):

$$\begin{array}{ccc}\mathrm{Ban}\times \mathrm{Ban}& \stackrel{\chi }{\to }& \mathrm{NVect}\\ \downarrow & & \downarrow \\ \mathrm{Vect}\times \mathrm{Vect}& \underset{\otimes }{\to }& \mathrm{Vect}\end{array}$$\array { Ban \times Ban & \overset{\chi}\rightarrow & NVect \\ \downarrow & & \downarrow \\ Vect \times Vect & \underset{\otimes}\rightarrow & Vect }

A uniform cross norm is obviously desirable from the nPOV, but does it meet the analysts' needs for a cross norm? Yes:

Proposition

A uniform cross norm assigns a cross norm to any two Banach spaces.

The specific cross norms from the previous section qualify as much as possible:

Proposition

The projective and injective cross norms are uniform cross norms (and hence are in fact cross norms). The norm on the algebraic tensor product of two Hilbert spaces is also a cross norm.

As far as I can tell, the Hilbert-space cross norm $\sigma$ doesn't apply to arbitrary Banach spaces, so it doesn't define a uniform cross norm as defined above; however, it does define a functor on $\mathrm{Hilb}\times \mathrm{Hilb}$, so it's as uniform as could be expected.

Looking only at the general theory of cross norms, the projective and injective cross norms appear naturally:

Proposition

If $\chi$ is any uniform cross norm, $V$ and $W$ are any Banach spaces, and $x$ is any element of $V\otimes W$, then

$$\parallel x{\parallel }_{ϵ}\le \parallel x{\parallel }_{\chi }\le \parallel x{\parallel }_{\pi }.$${\|x\|_\epsilon} \leq {\|x\|_\chi} \leq {\|x\|_\pi} .

That is, we have a poset of uniform cross norms, and the projective and injective cross norms are (respectively) the top and bottom of this poset.

Although $\sigma$ is not a uniform cross norm, it relates to $ϵ$ and $\pi$ in the same way:

Proposition

If $V$ and $W$ are Hilbert spaces and $x$ is an element of $V\otimes W$, then

$$\parallel x{\parallel }_{ϵ}\le \parallel x{\parallel }_{\sigma }\le \parallel x{\parallel }_{\pi }.$${\|x\|_\epsilon} \leq {\|x\|_\sigma} \leq {\|x\|_\pi} .

Actually, this would all be simpler if Propostion 3 applied to arbitrary cross norms and not just uniform ones. Perhaps it does. Or perhaps $\sigma$ extends to a uniform cross norm on all of $\mathrm{Ban}$; that would also make things simpler. I don't know.

Of course, any cross norm $\chi$ on $V$ and $W$ allows us to form the Banach space $V{\displaystyle {\hat{\otimes }}_{\chi }}W$, which may reasonably be called a tensor product of $V$ and $W$; that's why we care.

Foundational issues

We need the Hahn–Banach theorem for $ϵ$ to be a cross norm; but $\sigma$ and $\pi$ work regardless. Possibly some of the other propositions rely on some other form of the axiom of choice; I haven't seen their proofs.

References

I got pretty much all of my facts from Wikipedia:

• Topological tensor product (which also discusses tensor products of locally convex spaces),
• Tensor product of Hilbert spaces.