A topological vector space is a real or complex vector space which is endowed with a topology such that vector addition and scalar multiplication are continuous.
The most common example is a normed vector space, but there are topological vector spaces which are not normed vector spaces.
Here is an example: consider the set of infinitely differentiable functions which are defined on a open set D in Rn such that each function has compact support (i.e. to each function there exists a compact set so that the function is zero outside that compact set). We first define a collection of semi-norms, and the topology will then be defined as the coarsest topology which refines the topology defined by each of the norms. For a compact set K and a multi-index m = (m1, ..., mn) we define the (K, m) semi-norm to be the supremum of the differentiation first by x1 m1 times, then by x2 m2 times and so on K.
A topological vector space defined by a collection of semi-norms is called a Fréchet space if it is complete.
A topological vector space has a dual - the set of all continuous functionals, i.e. continuous linear maps from the space into the base field K (either R or C). The topology on the dual is defined to be the coarsest topology such that the appliance mapping V × V* -> K is continuous. This turns the dual into a topological vector space. Note that in the case of a Banach space, this gives the dual the weak topology.