In mathematics, the injective tensor product is a particular topological tensor product, a topological vector space (TVS) formed by equipping the tensor product of the underlying vector spaces of two TVSs with a compatible topology.
It was introduced by Alexander Grothendieck and used by him to define nuclear spaces.
Injective tensor products have applications outside of nuclear spaces: as described below, many constructions of TVSs, and in particular Banach spaces, as spaces of functions or sequences amount to injective tensor products of simpler spaces.
be locally convex topological vector spaces over
, with continuous dual spaces
Although written in terms of complex TVSs, results described generally also apply to the real case.
is isomorphic to the (vector space) tensor product
denote the respective dual spaces with the topology of bounded convergence.
is a locally convex topological vector space, then
, whose basic open sets are constructed as follows.
to form a locally convex TVS topology on
[1][clarification needed] This topology is called the
as above, and the topological vector space consisting of
Its norm can be expressed in terms of the (continuous) duals of
Denoting the unit balls of the dual spaces
under the injective norm is isomorphic as a topological vector space to
are two linear maps between locally convex spaces.
are continuous then so is their tensor product
-topology is the finest locally convex topology on
that makes continuous the canonical map
and called the projective tensor product of
[5] The injective and projective topologies both figure in Grothendieck's definition of nuclear spaces.
, a term justified by the following fact.
consists of exactly those continuous bilinear forms
running through a norm bounded subset of the space of Radon measures on
a Banach space, certain constructions related to
in Banach space theory can be realized as injective tensor products.
be the space of unconditionally summable sequences in
denotes the Banach space of continuous functions on
be a complete, Hausdorff, locally convex topological vector space, and let