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.
