Topological tensor product

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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, but for general Banach spaces or locally convex topological vector space the theory is notoriously subtle, and most mathematicians prefer to avoid it if possible.^{[citation needed]}

1 Tensor products of Hilbert spaces
2 Cross norms and tensor products of Banach spaces
3 Tensor products of locally convex topological vector spaces
4 See also
5 References

[edit] Tensor products of Hilbert spaces

The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form 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.

[edit] Cross norms and tensor products of Banach spaces

The obvious way to define the tensor product of two Banach spaces A and B 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.

A cross norm p on the algebraic tensor product of A and B is a norm satisfying the conditions

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

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

Here a′ and b′ are in the duals of A and B, and p′ is the dual norm of p. The term reasonable crossnorm is also used for the definition above.

There is a smallest cross norm λ called the injective cross norm,given by

$\lambda(x) = \sup |a'\otimes b'(x)|$

where the sup is taken over all pairs a′ and b′ of norm at most 1, and a largest cross norm γ called the projective cross norm, given by

$\gamma(x) = \inf \Sigma \|a_i\| \|b_i\|$

where the $\inf$ is taken over all finite decompositions $x = \Sigma a_i \otimes b_i$ .

The completions of the algebraic tensor product in these two norms are called the injective and projective tensor products, and are denoted by $A \otimes_\lambda B$ and $A \otimes_\gamma B$ .

The norm used for the Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by σ, so the Hilbert space tensor product in the section above would be $A \otimes_\sigma B$ .

A uniform crossnorm α is an assignment to each pair $(X, Y)$ of Banach spaces of a reasonable crossnorm on $X \otimes Y$ so that if $X$ , $W$ , $Y$ , $Z$ are arbitrary Banach spaces then for all (continuous linear) operators $S: X \to W$ and $T: Y \to Z$ the operator $S \otimes T : X \otimes_\alpha Y \to W \otimes_\alpha Z$ is continuous and $\|S \otimes T\| \leq \|S\| \|T\|$ (Ryan 2002).

A uniform crossnorm $α$ is said to be finitely generated if, for every pair $(X, Y)$ of Banach spaces and every $u \in X \otimes Y$ ,

$\alpha(u; X \otimes Y) = \inf \{ \alpha(u ; M \otimes N) : \dim M, \dim N < \infty \}.$

A tensor norm is defined to be a finitely generated uniform crossnorm.

A uniform crossnorm $α$ is cofinitely generated if, for every pair $(X, Y)$ of Banach spaces and every $u \in X \otimes Y$ ,

$\alpha(u) = \sup \{ \alpha((Q_E \otimes Q_F)u; (X/E) \otimes (Y/F)) : \dim X/E, \dim Y/F < \infty \}.$

[edit] Tensor products of locally convex topological vector spaces

The topologies of locally convex topological vector spaces A and B are given by families of seminorms. For each choice of seminorm on A and on B 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⊗_γB and A⊗_λB. There is a natural map from A⊗_γB to A⊗_λB.

If A or B is a nuclear space then the natural map from A⊗_γB to A⊗_λB is an isomorphism. Roughly speaking, this means that if A or B is nuclear, then there is only one sensible tensor product of A and B. This property characterizes nuclear spaces.

[edit] See also

[edit] References

Ryan, R.A. (2002), Introduction to Tensor Products of Banach Spaces, New York: Springer .
Grothendieck, A. (1955), “Produits tensoriels topologiques et espaces nucléaires”, Memoirs of the American Mathematical Society 16 .