More generally, an integral bilinear form is a bilinear functional that belongs to the continuous dual space of
, the injective tensor product of the locally convex topological vector spaces (TVSs) X and Y.
An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form.
denote the projective tensor product,
denote the injective tensor product, and
This identifies the continuous dual space of
denote its transpose, which is a continuous injection.
can be canonically identified as a vector subspace of
are called integral (bilinear) forms on
The following theorem justifies the word integral.
consists of exactly of the continuous bilinear forms u on
is a (necessarily bounded) positive Radon measure on the (compact) set
There is also a closely related formulation [3] of the theorem above that can also be used to explain the terminology integral bilinear form: a continuous bilinear form
equipped with a (necessarily bounded) positive Radon measure
can be realised by integrating (essentially bounded) functions on a compact space.
is of the form:[4] for suitable weakly closed and equicontinuous subsets S and T of
, one can define a canonical bilinear form
: for suitable weakly closed and equicontinuous aubsets
The following result shows that integral maps "factor through" Hilbert spaces.
is an integral map between locally convex TVS with Y Hausdorff and complete.
There exists a Hilbert space H and two continuous linear mappings
Furthermore, every integral operator between two Hilbert spaces is nuclear.
[6] Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.
[5] An important partial converse is that every integral operator between two Hilbert spaces is nuclear.
[6] Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that
is a continuous linear operator between two normed space then
is a continuous linear map between locally convex TVSs.
[5] Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete.
are all integral linear maps then their composition
[6] Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection