The finest locally convex topological vector space (TVS) topology on
the tensor product of two locally convex TVSs, making the canonical map
(defined by sending
) separately continuous is called the inductive topology or the
is endowed with this topology then it is denoted by
and called the inductive tensor product of
be locally convex topological vector spaces and
be a linear map.
is a locally convex space and that
is the canonical map from the space of all bilinear mappings of the form
going into the space of all linear mappings of
is restricted to
(the space of separately continuous bilinear maps) then the range of this restriction is the space
of continuous linear operators
In particular, the continuous dual space of
is canonically isomorphic to the space
the space of separately continuous bilinear forms on
τ
is a locally convex TVS topology on
with this topology will be denoted by
is equal to the inductive tensor product topology if and only if it has the following property:[5]