MAT454H1 Lecture Notes - Compact Space, Maximal And Minimal Elements, Normed Vector Space
Document Summary
Suppose x is a real vector space and u is a convex subset containing 0. r>0 ru = x. Given such a u de ne p = pu by p(x) = inf{r 0 : x ru}. p is called the gauge of u. Proposition 1. p is a positive minkowksi functional. If u = u it is a semi-norm. That p(ax) = ap(x) for a > 0 is clear. Proof: given x and y in x we may write x = (p(x) + )u and y = (p(y) + )v with u, v u. Thus x + y = (p(x) + + p(y) + )w, where w = 1 p(x) + + p(y) + (p(x) + )u + (p(y) + )v) u. Since > 0 is arbitrary we get subaddivity. Finally if u = u we get p(x) = p( x) so p is a semi-norm.
