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.
