Locally Convex Topological Vector Space
In
functional analysis
, a
topological vector space
is called
locally convex
if its
topology
is defined by a set of
convex
neighborhoods
of 0. Every
normed space
is locally convex, since the
triangle inequality
ensures that all
balls
are convex. More formally, a
locally convex topological vector space
(or
locally convex space
) is a
topological vector space
with the following
local convexity
condition: there exists a
base
of
neighbourhoods
of 0 consisting of convex sets. Equivalently, the topology is that defined by a family of
semi-norms
. Although such a space need not be
Hausdorff
, this is often also assumed. Every
Banach space
is a locally convex space, and much of the theory of locally convex spaces generalises parts of the theory of Banach spaces. Indeed, local convexity is a generalisation of
normable
strong enough for the
Hahn-Banach theorem
to hold, giving a sufficiently rich theory of continuous linear functionals. Many examples of locally convex topological vector spaces are described in the
topological vector space
article. On the other hand,
L
p
spaces
for
0
are not locally convex.
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