TAILIEUCHUNG - Ebook Problems in Real Analysis (Second Edition): Part 2

(BQ) Problems in Real Analysis teaches the basic methods of proof and problem-solving by presenting the complete solutions to over 600 problems that appear in Principles of Real Analysis, Third Edition. The problems are distributed in forty sections, and cover the entire spectrum of difficulty. | CHAPTER 5 NORMED SPACES AND p-SPACES 27. NORMED SPACES AND BANACH SPACES Problem . Let X be a normed space. Then show that X is a Banach space if and only if its unit sphere x X x 1 is a complete metric space under the Induced metric d x y Ị x y . Solution. Let s x s X x j 1 and note that 5 is a closed set. Clearly if X is a Banach space then s is a complete metric space. Conversely assume that 5 ỈS complete. Let xj be a Cauchy sequence of X In view of the inequality Ị IM - IMI I h 1 - m we see that IIxn II is a Cauchy sequence of real numbers. If lỉm II xn II 0 then limx 0. So we can assume that 5 ỉím ị xM II 0. In this case we can also assume that there exists some M 0 such that I M and IIxn II M both hold for each n. The inequalities II -V II il hwUn- .tnii-Tw O II Ikmlỉ ll IknlHtarll hd II xm x xra x II xm Qxm II 2M3 x xw show that the sequence I I is a Cauchy sequence of s. If X is its limit in s then x IIx II 5x holds in X and so X is a Banach space. IMn it Problem . Let X be a normed vector space. Fix a X and a nonzero scalar a. 239 240 Chapter S NORMED SPACES AND Lp -SPACES a. Show that the mappings X w- a X and X ax are both homeomorphisms. b. If A and B are two sets with either A or B open and a and f are nonzero scalars then show that fB is an open set. Solution a Observe that II a x a -r y x yII holds for all X and y. This shows that X a X is in fact an isometry. Also notice that for all X y e X we have j ax ay II dfI x y II. This easily implies that X H ax is a homeomorphism. b Assume first that B is an open set. Since the mapping X a X is a homeomorphism we know that a 4- B is an open set for each a X. This implies that the set A B 5 is an Pen set for each subset A of X. Now assume that B is an open set and that a and jỡ are nonzero scalars. Since the mapping X fix is a homeomorphism the set fB is an open set. So by the preceding discussion a A 4- fB must be an open set. Problem . Let X be a normed vector space and let B x X IIXII 1Ị be .

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