Chapter 2 Banach Space Definition 2.0.1. Let (X; d) and (Y; ρ) be two metric spaces. Let x0 be a point in X.A function f :(X; d) −! (Y; ρ) is said to be continuous at a point x0 2 X, if for an arbitrary chosen > 0, there exist a number δ > 0 such that ρ (f(x); f(x0)) < whenever d(x; x0) < δ Definition 2.0.2 (Sequential condition for continuity ). Let f :(X; d) −! (Y; ρ) be a 1 function. The function f is said to be continuous at a point x 2 X, iff every sequence (xn)n=1 1 in X converging to x,(f(xn))n=1 converges to f(x) in Y. 1 Definition 2.0.3 (Cauchy Sequence ). Let (X; d) be a metric space. A sequence (xn)n=1 in X is said to be a Cauchy sequence if for every > 0, there exist a natural number N = N() such that d(xm; xn) < for all m; n ≥ N. Theorem 2.0.4. Every convergent sequence in a metric space is a Cauchy sequence, but con- verse is not necessarily true. Definition 2.0.5 (Complete metric space ). Let (X; d) be a metric space. The metric space (X; d) is said to be a complete metric space, if every Cauchy sequence in X converges to some point in X. Theorem 2.0.6. For a metric space X = (X; d) there exists a complete metric space X^ = (X;^ d) which has a subspace W that is isometric with X and is dense in X^. This space X^ is unique except for isometries, that is, if X~ is any complete metric space having a dense subspace W~ isometric with X, then X^ and X~ are isometric. 8 CHAPTER 2. BANACH SPACE Definition 2.0.7 (Banach Space). A normed linear space (X; k · k) is said to be a Banach space if every Cauchy sequence in (X; k · k) converges to some point in (X; k · k). Example 2.0.8. (R; k · k) is a Banach space. Example 2.0.9. (C; k · k) is a Banach space. p Example 2.0.10. Show that ln; 1 ≤ p ≤ 1 space is a Banach space. Example 2.0.11. Show that lp; 1 ≤ p < 1 space is a Banach space. Example 2.0.12. Show that l1 space is a Banach space. Example 2.0.13. Let C denotes the vector space of all convergent sequences of complex num- bers. Define, 1 kxk1 = sup jxnj, x = (xn)n=1 2 C. n≥1 Show that (C; k · k1) is a Banach space. Example 2.0.14. Show that (C0; k · k1) is a Banach space. Example 2.0.15. Show that (CC[a; b]; k · k1) is not a Banach space. Theorem 2.0.16 (Completion). Let (X; k · k) be a normed linear space. Then there is a Banach space X^ and an isometry A from X onto a subspace W of X^ which is dense in X^. The space X^ is unique, except for isometries. Theorem 2.0.17 (Complete subspace). A subspace M of a complete metric space X is itself complete if and only if the set M is closed in X. Theorem 2.0.18. A normed linear space (X; k · k) is complete(or a Banach space) iff the unit sphere S = fx 2 X : kxk = 1g is complete. 1 1 Definition 2.0.19. Let (xn)n=1 be a sequence and (nk)k=1 be a strictly monotonic increasing 1 1 sequence of natural numbers, then (xnk )k=1 is said to be a subsequence of (xn)n=1. 1 Theorem 2.0.20. Let (X; d) be a metric space and (xn)n=1 be a Cauchy sequence in (X; d). 1 1 If a subsequence (xnk )k=1 of a Cauchy sequence (xn)n=1 converges to a point x in X, then the 1 original sequence (xn)n=1 also converges to same point x in X. 9 CHAPTER 2. BANACH SPACE 1 X Definition 2.0.21 (Absolutely convergent). The series xn is said to be absolutely con- n=1 1 X vergent, if the series kxnk of non-negative real numbers is convergent. n=1 Remark 2.0.22. An absolutely convergent series of complex numbers is convergent. Theorem 2.0.23. A normed linear space (X; k · k) is a Banach space iff every absolutely con- vergent series in X is convergent in X itself. Definition 2.0.24. Let (X; d) be a metric space and A be a subset of X. (I) A point x 2 X is said to be a limit point/accumulation point/cluster point of A if every open sphere S(x; r) with centre x and radius r > 0 contains at least one point of A other than x. (II) A is called closed if it contains all of its limit points. (III) Closure of A, denoted by A¯ is the intersection of all closed subsets of X containing A. i.e. A=¯ TfF: F is closed subset of X and A ⊂ F g Definition 2.0.25. Let X be a vector space over a field K(= R or C) and M be a subspace of X. Define X = fM + x : x 2 Xg M X Then M is a vector space under the following operation:- (i)( M + x) + (M + y) = M + (x + y) 8x; y 2 X (ii) α(M + x) = M + αx 8x 2 X; 8α 2 K X Remark 2.0.26. M is called a Quotient space. Theorem 2.0.27. Let (X; k k) be a normed linear space over K(= R or C) and M be a closed X subspace of X. Then M is a normed linear space under a norm defined by kM + xk = inf kx + mk = inf kx − mk. m2M m2M Theorem 2.0.28. Let (X; k k) be a normed linear space and M be a closed subspace of X. X The normed linear space M is a Banach space if X is a Banach space. 10 CHAPTER 2. BANACH SPACE Theorem 2.0.29. Every finite dimensional normed linear space X over a field K(= R or C) is complete (or X is a Banach space). Corollary 2.0.30. Every finite dimensional subspace of a normed linear space X is closed in X. Definition 2.0.31 (Compactness). A metric space X is said to be compact (sequentially compact) if every sequence in X has a convergent subsequence. Definition 2.0.32. A subset A of X is said to be compact if every sequence in A has a convergent subsequence whose limit is an element of A. Corollary 2.0.33. Every closed and bounded subset of a finite dimensional normed linear space X is compact. Definition 2.0.34 (Equivalent norms). Let X be a vector space over a field K(= R or C). Let k · k1 and k · k2 be two norms on X. A norm k · k1 on X is said to be equivalent to a norm k · k2 if there are positive numbers a and b such that for all x 2 X we have akxk2 ≤ kxk1 ≤ bkxk2 n Example 2.0.35. Prove that in R , k · k1 and k · k1 are equivalent. n Example 2.0.36. Prove that in R , k · k1 and k · k2 are equivalent. Remark 2.0.37. Equivalent norms on X define the same topology for X. Theorem 2.0.38. On a finite dimensional vector space X, any norm k · k1 is equivalent to any other norm k · k2. Lemma 2.0.39 (Riesz's Lemma). Let Y and Z be subspaces of a normed space X (of any dimension), and suppose that Y is closed and is a proper subset of Z. Then for every real number θ in the interval (0; 1) there is a z 2 Z such that kzk = 1; kz − yk ≥ θ 8 y 2 Y: Theorem 2.0.40. The closed unit ball in a normed linear space X over a field K(= R or C) is compact iff X is finite dimensional. THE END 11.