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
as being identical to 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.
[6] Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection