One of the original motivations for topological tensor products
is the fact that tensor products of the spaces of smooth real-valued functions on
cannot be expressed as a finite linear combination of smooth functions in
[1] 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.
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.
The obvious way to define the tensor product of two Banach spaces
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.
are Banach spaces the algebraic tensor product of
are Banach spaces, a crossnorm (or cross norm)
are elements of the topological dual spaces of
The term reasonable crossnorm is also used for the definition above.
It turns out that the projective cross norm agrees with the largest cross norm ((Ryan 2002), pp.
Note hereby that the injective cross norm is only in some reasonable sense the "smallest".
The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by
are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general.
so the Hilbert space tensor product in the section above would be
of Banach spaces of a reasonable crossnorm on
are arbitrary Banach spaces then for all (continuous linear) operators
defines a reasonable cross norm on the algebraic tensor product
The normed linear space obtained by equipping
A tensor norm is defined to be a finitely generated uniform crossnorm.
we can define the corresponding family of cross norms on the algebraic tensor product
There are in general an enormous number of ways to do this.
are called the projective and injective tensor products, and denoted by
is a nuclear space then the natural map from
This property characterizes nuclear spaces.