Topological Tensor Product - Tensor Products of Locally Convex Topological Vector Spaces

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.

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