Explore BrainMass

Norms and Bounded Sets

8. Fix an n-dimensional real vector space V with n a positive integer greater than 1. If you want to take V to be R, fine. Consider non-empty open sets B C V with the following properties:
(a) B is bounded and convex (contains the line segment through any two of its
(b) If VEB,then there is a number t0>0 for which tv EB......
(i) If ||.|| is any norm on V, show that B={v cv: flvfl <1}has properties (a) and (b).
(ii) Conversely, suppose B is a non-empty open set with properties (a) and (b). Show that there exists a norm on V for which B is the open ball of radius I about 0 relative to ii. fl. Hint: With S =OB the boundary of B, show that for any non-zero w in F, there is a unique c>Ofor which v=cw E S and then define ... to be 1/c. Use properties (a) and (b) to showthatli'll isanorm.

Please see the attached file for the fully formatted problems.


Solution Summary

Norms and Bounded Sets are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.