Download SEMINORMS AND LOCAL CONVEXITY A family P of seminorms on

SEMINORMS AND LOCAL CONVEXITY
JOHN QUIGG
A family P of seminorms on X is separating if for all x 6= 0 there exists p ∈ P such that
p(x) 6= 0.
Let P be a separating family of seminorms on a vector space X. The topology generated by
P is the projective topology generated by the maps X → X/ ker p for p ∈ P.
A subset A of a vector space X is absorbing if for all x ∈ X there exists c > 0 such that
x ∈ tA for all t > c.
The Minkowski functional of an absorbing subset A of X is the function µA : X → R defined
by
µA (x) = inf{t > 0 : x ∈ tA}.
Proposition. If A is a convex balanced absorbing subset of X, then µA is a seminorm.
Conversely, let p be a seminorm on X, and put
B = {x : p(x) < 1}.
Then B is convex, balanced, and absorbing, and p = µB .
Theorem. On any vector space, the topology generated by a separating family P of seminorms is locally convex, and makes every p ∈ P continuous. Conversely, every locally convex
space X has topology generated by a family P of seminorms.
If X 0 is a separating vector space of linear functionals on a vector space X, then the projective
topology generated by X 0 is generated by the seminorms
for f ∈ X 0 .
x 7→ |f (x)|
Corollary. In a locally convex space, a set is bounded if and only if it is weakly bounded.
Theorem. A locally convex space is metrizable if and only if its topology is generated by a
countable family of seminorms.
Theorem. A topological vector space is normable if and only if it is locally bounded and
locally convex.
Date: September 6, 2005.
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