A topology on a topological vector space (with usually assumed to be T2) is said to be locally convex if admits a local base at consisting of balanced, convex, and absorbing sets. In some older literature, the definition of locally convex is often stated without requiring that the local base be balanced or absorbing.
It is not unusual to blur the distinction as to whether "locally convex" applies to the topology on or to itself.
The above definition can also be stated in terms of seminorms. In particular, a topological vector space (with assumed ) is locally convex if is generated by a family of seminorms satisfying
where denotes the zero vector in and is different from 0 which denotes the element 0 in the scalar field of . The condition (1) above ensures that is ; removal of this criterion on allows one to remove condition (1), whereby is locally convex if and only if is generated by a family of seminorms.
The seminorm condition illustrates why local convexity is a desirable property. In particular, topological vector spaces which are locally convex can be thought of as generalizations of normed spaces, thereby allowing considerable functional analysis to be done even without the existence of a norm.