Topological tensor product

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. One of the original motivations for topological tensor products is the fact that tensor products of the spaces of smooth functions on do not behave as expected. There is an injection but this is not an isomorphism. For example, the function cannot be expressed as a finite linear combination of smooth functions in We only get an isomorphism after constructing the topological tensor product; i.e., This article first details the construction in the Banach space case. is not a Banach space and further cases are discussed at the end. 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 ai and bj run through orthonormal bases of A and B, then the vectors ai⊗bj form an orthonormal basis of A ⊗ B. 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. If and are Banach spaces the algebraic tensor product of and means the tensor product of and as vector spaces and is denoted by The algebraic tensor product consists of all finite sums where is a natural number depending on and and for When and are Banach spaces, a (or ) on the algebraic tensor product is a norm satisfying the conditions Here and are elements of the topological dual spaces of and respectively, and is the dual norm of The term is also used for the definition above.

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Topological tensor product

Complete topological vector space

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces.

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

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