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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... Jump to: navigation, search In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... Jump to: navigation, search In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...

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 base of A⊗B.

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⊗b) = ||a|| ||b||

p′(a′⊗b′)= ||a′|| ||b′||

Here a′ and b′ are in the duals of A and B, and p′ is the dual norm of p.

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

λ(x) = sup |a′⊗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

γ(x)=inf Σ ||a_i|| ||b_i||

where the inf is taken over all finite decompositions x= Σa_i⊗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⊗_λB and A⊗_γB.

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⊗_σB.

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. Jump to: navigation, search In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. ...

See also

• Hilbert space, Banach space, Frechet space, locally convex topological vector space, Nuclear space
• tensor product
• Fredholm kernel

In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... Jump to: navigation, search In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... This article deals with Fréchet spaces in functional analysis. ... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... Jump to: navigation, search In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... Jump to: navigation, search In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. ...

References

• A. Grothendieck, "Produits tensoriels topologiques et espace nuclieares", (1955) Memoirs of the American Mathematical Society 16.