In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product.
Namely, given locally convex topological vector spaces
a locally convex topological vector space such that the canonical map
and called the projective tensor product of
be locally convex topological vector spaces.
is the unique locally convex topological vector space with underlying vector space
is the balanced convex hull of the set
is generated by the collection of such tensor products of the seminorms on
[3] Throughout, all spaces are assumed to be locally convex.
denotes the completion of the projective tensor product of
can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by
, namely, the space of continuous bilinear forms
[9] In a Hausdorff locally convex space
[10] The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.
be metrizable locally convex TVSs and let
is the sum of an absolutely convergent series
The next theorem shows that it is possible to make the representation of
) be a balanced open neighborhood of the origin in
be a compact subset of the convex balanced hull of
denote the families of all bounded subsets of
is the space of continuous bilinear forms
the topology of uniform convergence on sets in
which is also called the topology of bi-bounded convergence.
This is equivalent to the problem: Given a bounded subset
is a subset of the closed convex hull of
Grothendieck proved that these topologies are equal when
They are also equal when both spaces are Fréchet with one of them being nuclear.
be a locally convex topological vector space and let
be locally convex topological vector spaces with
Then, denoting strong dual spaces with a subscripted