Norm (mathematics) - Classification of Seminorms: Absolutely Convex Absorbing Sets

Classification of Seminorms: Absolutely Convex Absorbing Sets

All seminorms on a vector space V can be classified in terms of absolutely convex absorbing sets in V. To each such set, A, corresponds a seminorm p_A called the gauge of A, defined as

p_A(x) := inf{α : α > 0, x ∈ α A}

with the property that

{x : p_A(x) < 1} ⊆ A ⊆ {x : p_A(x) ≤ 1}.

Conversely:

Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct such a basis is to use a separating family (p) of seminorms p: the collection of all finite intersections of sets {p<1/n} turns the space into a locally convex topological vector space so that every p is continuous.

Such a method is used to design weak and weak* topologies.

norm case:

Suppose now that (p) contains a single p: since (p) is separating, p is a norm, and A={p<1} is its open unit ball. Then A is an absolutely convex bounded neighbourhood of 0, and p = p_A is continuous.

The converse is due to Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely:

If V is an absolutely convex bounded neighbourhood of 0, the gauge g_V (so that V={g_V <1}) is a norm.