Purchase Solution

Norms and Bounded Sets

Not what you're looking for?

Ask Custom Question

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.
.....

Please see the attached file for the fully formatted problems.

Attachments
Purchase this Solution

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.

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

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 ...

Purchase this Solution


Free BrainMass Quizzes
Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Probability Quiz

Some questions on probability

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.