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# 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
points);
(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.
.....

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Problem #8
Proof:
1. Since , then the border of is . We want to show that has properties (a) and (b).
(a) From the definition, we know that is bounded and the bound is 1. For any , any point on the line segment has the form for some . Then we have . Thus and hence the whole line segment is contained in . Therefore, is convex.
(b) For any , let , then for any , we ...

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