CHEM 151B Midterm: Chem 151B UCSC Exam 1 2016
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You have 60 minutes to answer all four questions. Let an be a sequence of nonempty closed subsets of a metric space (x; d) with an+1 (cid:26) an for all n. show that if one of the sets an is sequentially compact, then \ Hint: remember: a set a in a metric space (x; d) is sequentially compact if every sequence of elements of a contains a convergent subsequence whose limit lies in a. n2n. In this case, a = p dp t where the diagonal entries of d are the eigenvalues of a and the columns of p are the corresponding eigenvectors. Use this result to show that for any x in rn, (cid:21)min kxk2 (cid:20) q(x)(cid:20)(cid:21)max kxk2 where (cid:21)min and (cid:21)max are the smallest and largest eigenvalues of a respectively, k(cid:1)k denotes the norm, and q : rn (cid:0)! R is de(cid:133)ned as q(x) = xtax (the quadratic form given by a).
