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.