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 ∈ 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 ∞ function. The function f is said to be continuous at a point x ∈ X, iff every sequence (xn)n=1 ∞ in X converging to x,(f(xn))n=1 converges to f(x) in Y.

∞ 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 Xˆ = (X,ˆ d) which has a subspace W that is isometric with X and is dense in Xˆ. This space Xˆ is unique except for isometries, that is, if X˜ is any complete metric space having a dense subspace W˜ isometric with X, then Xˆ and X˜ are isometric.

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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 ≤ ∞ space is a Banach space.

Example 2.0.11. Show that lp, 1 ≤ p < ∞ space is a Banach space.

Example 2.0.12. Show that l∞ space is a Banach space.

Example 2.0.13. Let C denotes the vector space of all convergent sequences of complex num- bers. Define,

∞ kxk∞ = sup |xn|, x = (xn)n=1 ∈ C. n≥1

Show that (C, k · k∞) is a Banach space.

Example 2.0.14. Show that (C0, k · k∞) 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 Xˆ and an isometry A from X onto a subspace W of Xˆ which is dense in Xˆ. The space Xˆ 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 = {x ∈ X : kxk = 1} is complete.

∞ ∞ Definition 2.0.19. Let (xn)n=1 be a sequence and (nk)k=1 be a strictly monotonic increasing ∞ ∞ sequence of natural numbers, then (xnk )k=1 is said to be a subsequence of (xn)n=1.

∞ Theorem 2.0.20. Let (X, d) be a metric space and (xn)n=1 be a Cauchy sequence in (X, d). ∞ ∞ If a subsequence (xnk )k=1 of a Cauchy sequence (xn)n=1 converges to a point x in X, then the ∞ original sequence (xn)n=1 also converges to same point x in X.

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∞ X Definition 2.0.21 (Absolutely convergent). The series xn is said to be absolutely con- n=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 ∈ 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=¯ T{F: F is closed subset of X and A ⊂ F }

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 = {M + x : x ∈ X} M X Then M is a vector space under the following operation:-

(i)( M + x) + (M + y) = M + (x + y) ∀x, y ∈ X

(ii) α(M + x) = M + αx ∀x ∈ X, ∀α ∈ 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. m∈M m∈M 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.

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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 ∈ X we have

akxk2 ≤ kxk1 ≤ bkxk2

n Example 2.0.35. Prove that in R , k · k1 and k · k∞ 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 ∈ Z such that

kzk = 1, kz − yk ≥ θ ∀ y ∈ 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

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