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 pA called the gauge of A, defined as
- pA(x) := inf{α : α > 0, x ∈ α A}
with the property that
- {x : pA(x) < 1} ⊆ A ⊆ {x : pA(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 = pA 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 gV (so that V={gV <1}) is a norm.
Read more about this topic: Norm (mathematics)
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