(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

a) Let V be a normed vector space. Then show that (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.

b) Show that a closed subset F of 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.)

2. Relevant equations

3. The attempt at a solution

I proved part a, but I really have no idea how to do part b. Why do we need compactness, and what is the suitable compact subset of F?

I hope someone can help me out! Thank you!

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# Homework Help: Closed subset of R^n has an element of minimal norm

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