A topological vector space is locally convex if it has a base of topology consisting of convex open subsets. Equivalently, it is a vector space equipped with a gauge consisting of seminorms. As with other topological vector spaces, a locally convex space (LCS) is often assumed to be Hausdorff space.
Locally convex (topological vector) spaces are the standard setup for much of the contemporary functional analysis.
The collections of functionals on a LCTVS is used in a way analogous to the collection of coordinate projections . For example, curves in a LCTVS over the reals can be composed with functionals to arrive at a collection of functions which are analogous to the ‘components’ of the curve.
In one respect, a locally convex TVS is a nice space? in that there are enough co-probes by maps to the base field.
Diagram of properties
J. L. Taylor, Notes on locally convex topological vector spaces (1995) (pdf)