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 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.
Let and be Banach spaces, and let be their tensor product as vector spaces. To define a tensor product of and as Banach spaces, we will place a norm on , making a normed vector space; the only difference in the following definitions is which norm to use. Then we take the completion , which is a Banach space.
Every element of may be written (in many different ways) as a formal linear combination of formal tensor products of elements of and (suppressing the symbol ):
Let the projective cross norm of an element of be
Then the projective tensor product of and is the completion of under the projective cross norm.
If and are linear functionals on and (respectively), then is a linear functional on . Let the injective cross norm of an element of be
Then the injective tensor product of and is the completion of under the injective cross norm.
and extend by linearity. We write the norm of an element of the inner product space as . Then the tensor product of the Hilbert spaces and is the completion of under this norm (or inner product).
Besides the specific norms defined above, we can define axioms of a reasonable norm on .
A cross norm on and is any norm on such that:
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 that makes the following diagram commute (or fills it with a natural isomorphism):
A uniform cross norm is obviously desirable from the nPOV, but does it meet the analysts' needs for a cross norm? Yes:
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:
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 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 , 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:
If is any uniform cross norm, and are any Banach spaces, and is any element of , then
Although is not a uniform cross norm, it relates to and in the same way:
If and are Hilbert spaces and is an element of , then
Actually, this would all be simpler if Propostion 3 applied to arbitrary cross norms and not just uniform ones. Perhaps it does. Or perhaps extends to a uniform cross norm on all of ; that would also make things simpler. I don't know.
Of course, any cross norm on and allows us to form the Banach space , which may reasonably be called a tensor product of and ; that's why we care.
I got pretty much all of my facts from Wikipedia: