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. 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 ⊗ ^ {displaystyle {hat {otimes }}} is the fact that tensor products of the spaces of smooth functions on R n {displaystyle mathbb {R} ^{n}} do not behave as expected. There is an injection but this is not an isomorphism. For example, the function f ( x , y ) = e x y {displaystyle f(x,y)=e^{xy}} cannot be expressed as a finite linear combination of smooth functions in C ∞ ( R x ) ⊗ C ∞ ( R y ) . {displaystyle C^{infty }(mathbb {R} _{x})otimes C^{infty }(mathbb {R} _{y}).} We only get an isomorphism after constructing the topological tensor product; i.e., This article first details the construction in the Banach space case. C ∞ ( R n ) {displaystyle C^{infty }(mathbb {R} ^{n})} is not a Banach space and further cases are discussed at the end. 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 (Ryan 2002) in this section. 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. If A and B are Banach spaces the algebraic tensor product of A and B means the tensor product of A and B as vector spaces and is denoted by A ⊗ B {displaystyle Aotimes B} . The algebraic tensor product A ⊗ B {displaystyle Aotimes B} consists of all finite sums where n {displaystyle n} is a natural number depending on x {displaystyle x} and a i ∈ A {displaystyle a_{i}in A} and b i ∈ B {displaystyle b_{i}in B} for i = 1 , … , n {displaystyle i=1,ldots ,n} .