標題: Real analysis: norms [打印本頁] 作者: kingwinner 時間: 10-4-12 06:55 PM
標題: Real analysis: norms
"Let V be a normed vector space. Then by the triangle inequality, the function f(x)=||x|| is a Lipschitz function from V into [0,∞). In particular, f is uniformly continuous on V."
Show that a closed subset F of R^n contains an element of minimal norm, that is, there is an x E F such that ||x||≤||y|| for all y E F. (here ||x|| refers to the usual Euclidean norm).
(hint: F may not be compact, so work on a suitable compact subset of F.)