A vector space with a T2-space topology such that the operations of vector addition and scalar multiplication are continuous. The interesting examples are infinite-dimensional spaces, such as a space of functions. For example, a Hilbert space and a Banach space are topological vector spaces.
The choice of topology reflects what is meant by convergence of functions. For instance, for functions whose integrals converge, the Banach space ,
one of the L-p-spaces, is used. But if one is
interested in pointwise convergence, then
no norm will suffice. Instead, for each
define the seminorm
on the vector space of functions on . The seminorms define a topology, the smallest one in which
the seminorms are continuous. So
is equivalent to
for all
, i.e., pointwise
convergence. In a similar way, it is possible to define a topology for which
"convergence" means uniform convergence
on compact sets.