Topological tensor product explained

In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

Motivation

One of the original motivations for topological tensor products

\hat{ ⊗ }

is the fact that tensor products of the spaces of smooth real-valued functions on

\Rⁿ

do not behave as expected. There is an injection

C^infty(\Rⁿ⁾ ⊗ C^infty(\R^m)\hookrightarrowC^infty(\R^n+m)

but this is not an isomorphism. For example, the function

f(x,y)=e^xy

cannot be expressed as a finite linear combination of smooth functions in

	infty(\R
C
	x) ⊗

	infty(\R
C
	y).

^[1] We only get an isomorphism after constructing the topological tensor product; i.e.,

C^infty(\Rⁿ⁾\hat{ ⊗

} C^\infty(\R^m) \cong C^\infty(\R^).

This article first details the construction in the Banach space case. The space

C^infty(\Rⁿ⁾

is not a Banach space and further cases are discussed at the end.

Tensor products of Hilbert spaces

See main article: Tensor product of Hilbert spaces. The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A ⊗ B, called the (Hilbert space) tensor product of A and B.

If the vectors a_i and b_j run through orthonormal bases of A and B, then the vectors a_i⊗b_j form an orthonormal basis of A ⊗ B.

Cross norms and tensor products of Banach spaces

We shall use the notation from in this section. The obvious way to define the tensor product of two Banach spaces

and

is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.

and

are Banach spaces the algebraic tensor product of

and

means the tensor product of

and

as vector spaces and is denoted by

A ⊗ B.

The algebraic tensor product

A ⊗ B

consists of all finite sums

x = \sum_^n a_i \otimes b_i,

where

is a natural number depending on

and

a_i\inA

and

b_i\inB

for

i=1,\ldots,n.

When

and

are Banach spaces, a (or )

on the algebraic tensor product

A ⊗ B

is a norm satisfying the conditions

p(a \otimes b) = \|a\| \|b\|,

p'(a' \otimes b') = \|a'\| \|b'\|.

Here

a^\prime

and

b^\prime

are elements of the topological dual spaces of

and

respectively, and

p^\prime

is the dual norm of

The term is also used for the definition above.

There is a cross norm

\pi

called the projective cross norm, given by

\pi(x) = \inf \left\,

where

x\inA ⊗ B.

It turns out that the projective cross norm agrees with the largest cross norm (pp. 15-16).

There is a cross norm

\varepsilon

called the injective cross norm, given by

\varepsilon(x) = \sup \left\

where

x\inA ⊗ B.

Here

A^\prime

and

B^\prime

denote the topological duals of

and

respectively.

Note hereby that the injective cross norm is only in some reasonable sense the "smallest".

The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by

A\operatorname{\hat{ ⊗ }}_\piB

and

A\operatorname{\hat{ ⊗ }}_\varepsilonB.

When

and

are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by

\sigma,

so the Hilbert space tensor product in the section above would be

A\operatorname{\hat{ ⊗ }}_\sigmaB.

\alpha

is an assignment to each pair

(X,Y)

of Banach spaces of a reasonable crossnorm on

X ⊗ Y

so that if

X,W,Y,Z

are arbitrary Banach spaces then for all (continuous linear) operators

S:X\toW

and

T:Y\toZ

the operator

S ⊗ T:X ⊗ _\alphaY\toW ⊗ _\alphaZ

is continuous and

\|S ⊗ T\|\leq\|S\|\|T\|.

and

are two Banach spaces and

\alpha

is a uniform cross norm then

\alpha

defines a reasonable cross norm on the algebraic tensor product

A ⊗ B.

The normed linear space obtained by equipping

A ⊗ B

with that norm is denoted by

A ⊗ _\alphaB.

The completion of

A ⊗ _\alphaB,

which is a Banach space, is denoted by

A\operatorname{\hat{ ⊗ }}_\alphaB.

The value of the norm given by

\alpha

A ⊗ B

and on the completed tensor product

A\operatorname{\hat{ ⊗ }}_\alphaB

for an element

A\operatorname{\hat{ ⊗ }}_\alphaB

(or

A ⊗ _\alphaB

) is denoted by

\alpha_A,B(x)or\alpha(x).

A uniform crossnorm

\alpha

is said to be if, for every pair

(X,Y)

of Banach spaces and every

u\inX ⊗ Y,

\alpha(u; X \otimes Y) = \inf \.

A uniform crossnorm

\alpha

is if, for every pair

(X,Y)

of Banach spaces and every

u\inX ⊗ Y,

\alpha(u) = \sup \.

A is defined to be a finitely generated uniform crossnorm. The projective cross norm

\pi

and the injective cross norm

\varepsilon

defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively.

and

are arbitrary Banach spaces and

\alpha

is an arbitrary uniform cross norm then

\varepsilon_(x) \leq \alpha_(x) \leq \pi_(x).

Tensor products of locally convex topological vector spaces

The topologies of locally convex topological vector spaces

and

are given by families of seminorms. For each choice of seminorm on

and on

we can define the corresponding family of cross norms on the algebraic tensor product

A ⊗ B,

and by choosing one cross norm from each family we get some cross norms on

A ⊗ B,

defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on

A ⊗ B

are called the projective and injective tensor products, and denoted by

A ⊗ _\gammaB

and

A ⊗ _λB.

There is a natural map from

A ⊗ _\gammaB

A ⊗ _λB.

is a nuclear space then the natural map from

A ⊗ _\gammaB

A ⊗ _λB

is an isomorphism. Roughly speaking, this means that if

is nuclear, then there is only one sensible tensor product of

and

.This property characterizes nuclear spaces.

References

Notes and References

Web site: What is an example of a smooth function in C∞(R2) which is not contained in C∞(R)⊗C∞(R) .