Math 3220
Introduction to Analysis
Summer term, 2000

Homework Assignments

Send e-mail to : Professor Korevaar
or to our grader/tutor Darrell Poore

Links:
Math 3220 homework page
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Professor Korevaar's home page
Department of Mathematics




Homework due Friday May 26

I) Use the three norm axioms on page 1 of Friday May 19 class notes, (i.e. statements i,ii,iii from Theorem 8.5 page 226 Wade 2nd edition), to show that if ||*|| is a norm, then d(x,y):= ||x-y|| defines a distance function, i.e.
(i) d(x,y) is always non-negative and can equal 0 iff x=y
(ii) d(x,y)=d(y,x), for all x,y
(iii) d(x,z) is less than or equal to d(x,y)+d(y,z), for all x,y,z.

II) For each of the four inequalities proven in the theorem on page 2 of Friday's notes, find vectors for which the inequalities become equalities. The inequalities are:
1a): (1/sqrt(n))*||x||_1 <= ||x||_2.
1b) ||x||_2 <= ||x||_1.
2a) ||x||_(infty) <= ||x||_2.
2b) ||x||_2 <= sqrt(n)*||x||_(infty).

%8.2 2nd edition, page 236: #2-7

Grading: 23 points possible, distributed as follows: I: 3 points; II:4 points; 2a,b,c: 1 point each; 3: 2 points; 4: 1 point; 5a,b: 2 points each; 6: 1 point for each of 4 parts; 7: 2 points.