Multivariable Advanced Calculus

82 SEQUENCES10. Suppose A ⊆ R n and z ∈ co (A) . Thus z = ∑ pk=1 t ka k for t k ≥ 0 and ∑ k t k =1. Show there exists n + 1 of the points {a 1 , · · · , a p } such that z is a convexcombination of these n + 1 points. Hint: Show that if p > n + 1 then the vectors{a k − a 1 } p k=2must be linearly dependent. Conclude from this the existence ofscalars {α i } such that ∑ pi=1 α ia i = 0. Now for s ∈ R, z = ∑ pk=1 (t k + sα i ) a k .Consider small s and adjust till one or more of the t k + sα k vanish. Now you arein the same situation as before but with only p−1 of the a k . Repeat the argumenttill you end up with only n + 1 at which time you can’t repeat again.11. Show that any uncountable set of points in F n must have a limit point.12. Let V be any finite dimensional vector space having a basis {v 1 , · · · , v n } . Forx ∈ V, letn∑x = x k v kk=1so that the scalars, x k are the components of x with respect to the given basis.Define for x, y ∈ Vn∑(x · y) ≡ x i y iShow this is a dot product for V satisfying all the axioms of a dot product presentedearlier.13. In the context of Problem 12 let |x| denote the norm of x which is produced bythis inner product and suppose ||·|| is some other norm on V . Thusi=1|x| ≡( ∑i|x i | 2 ) 1/2wherex = ∑ kx k v k . (4.8)Show there exist positive numbers δ < ∆ independent of x such thatδ |x| ≤ ||x|| ≤ ∆ |x|This is referred to by saying the two norms are equivalent. Hint: The top half iseasy using the Cauchy Schwarz inequality. The bottom half is somewhat harder.Argue that if it is not so, there exists a sequence {x k } such that |x k | = 1 butk −1 |x k | = k −1 ≥ ||x k || and then note the vector of components of x k is onS (0, 1) which was shown to be sequentially compact. Pass to a limit in 4.8 anduse the assumed inequality to get a contradiction to {v 1 , · · · , v n } being a basis.14. It was shown above that in F n , the sequentially compact sets are exactly thosewhich are closed and bounded. Show that in any finite dimensional normed vectorspace, V the closed and bounded sets are those which are sequentially compact.15. Two norms on a finite dimensional vector space, ||·|| 1and ||·|| 2are said to beequivalent if there exist positive numbers δ < ∆ such thatδ ||x|| 1≤ ||x|| 2≤ ∆ ||x 1 || 1.Show the statement that two norms are equivalent is an equivalence relation.Explain using the result of Problem 13 why any two norms on a finite dimensionalvector space are equivalent.