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 and , the projective topology, or π-topology, on is the strongest topology which makes a locally convex topological vector space such that the canonical map (from to ) is continuous. When equipped with this topology, is denoted and called the projective tensor product of and .
Let and be locally convex topological vector spaces. Their projective tensor product is the unique locally convex topological vector space with underlying vector space having the following universal property:[1]
When the topologies of and are induced by seminorms, the topology of is induced by seminorms constructed from those on and as follows. If is a seminorm on , and is a seminorm on , define their tensor product to be the seminorm on given by
Throughout, all spaces are assumed to be locally convex. The symbol denotes the completion of the projective tensor product of and .
In general, the space is not complete, even if both and are complete (in fact, if and are both infinite-dimensional Banach spaces then is necessarily not complete[8]). However, can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by .
The continuous dual space of is the same as that of , namely, the space of continuous bilinear forms .[9]
In a Hausdorff locally convex space a sequence in is absolutely convergent if for every continuous seminorm on [10] We write if the sequence of partial sums converges to in [10]
The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.[11]
Theorem — Let and be metrizable locally convex TVSs and let Then is the sum of an absolutely convergent series
The next theorem shows that it is possible to make the representation of independent of the sequences and
Theorem[12] — Let and be Fréchet spaces and let (resp. ) be a balanced open neighborhood of the origin in (resp. in ). Let be a compact subset of the convex balanced hull of There exists a compact subset of the unit ball in and sequences and contained in and respectively, converging to the origin such that for every there exists some such that
Let and denote the families of all bounded subsets of and respectively. Since the continuous dual space of is the space of continuous bilinear forms we can place on the topology of uniform convergence on sets in which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on , and in (Grothendieck 1955), Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset do there exist bounded subsets and such that is a subset of the closed convex hull of ?
Grothendieck proved that these topologies are equal when and are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck[13]). They are also equal when both spaces are Fréchet with one of them being nuclear.[9]
Let be a locally convex topological vector space and let be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:
Theorem[14] (Grothendieck) — Let and be locally convex topological vector spaces with nuclear. Assume that both and are Fréchet spaces, or else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted :