A locally convex topology 
on a topological
vector space 
(with 
usually assumed to be T2) is one that
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 T2)
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 
. This separation condition ensures that
is T2;
removal of this criterion on 
allows one to remove the separation condition, whereby 
is locally convex iff 
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.
