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CHAPTER 9 Inner Product Spaces and Hilbert Spaces Prof. M. Saha CHAPTER 9 Inner Product Spaces and Hilbert Spaces BY Prof. M. Saha Professor of Mathematics The University of Burdwan West Bengal, India E-mail : [email protected] Chapter 9 Inner Product Spaces and Hilbert Spaces Introduction and Objectives In the preceding chapters, we discussed normed linear spaces and Banach spaces. These spaces has linear properties as well as metric properties. Although the norm on a linear space generalizes the elementary concept of the length of a vector, but the main geometric concept other than the length of a vector is the angle between two vectors, In this chapter, we take the opportunity to study linear spaces having an inner product, a generalization of the usual dot product on finite dimensional linear spaces. The concept of an inner product in a linear space leads to an inner product space and a complete inner product space which is called a Hilbert space. The theory of Hilbert Spaces does not deal with angles in general. Most interestingly, it helps us to introduce an idea of perpendicularity for two vectors and the geometry deals in various fundamental aspects with Euclidean geometry. The basics of the theory of Hilbert spaces was given by in 1912 by the work of German mathematician D. Hilbert (1862 -1943) on integral equations. However, an axiomatic basis of the theory was given by famous mathematician J. Von Neumann (1903 -1957). However, Hilbert spaces are the simplest type of infinite dimensional Banach spaces to tackle a remarkable role in functional analysis. This chapter is divided into four modules. Module 1 ideals with the introduction of inner product spaces and Hilbert spaces with some examples. Some basic properties are given in it. Module 2 is concerned with the conception of orthogonal and orthonormal vectors in a Hilbert space. Some basic result are shown.. Module 3 is devoted mainly with very fundamental results on inner product spaces. Module 4 is discussed on some fundamental results on the Hilbert spaces particularly on approximation theory of convex subsets of a Hilbert spaces. Module 5 is emphasized on infinite series are those convergence in Hilbert spaces. Also with an introduction of isometric isomorphism property between two infinite di- mensional Hilbert spaces an important theorem namely Reisz Fischer's Theorem has been stated. 2 Module-1: Inner Product Spaces Definition 9.1.1: Let, X be a linear space over a field of complex numbers. If for every pair (x; y) 2 X × X there corresponds a scalar denoted by hx; yi called inner product of x and y of X such that the following properties hold. (IP.1) hx; yi = hy; xi where (x; y) 2 X × X and ` ' denotes the conjugate of the complex number. (IP.2) for all α 2 C; hαx; yi = αhx; yi; 8 (x; y) 2 X × X. (IP.3) for all x; y; z 2 X hx + y; zi = hx; zi + hy; zi. (IP.4) hx; xi ≥ 0 and hx; xi = 0 iff x = θ. Then (X; h : i) is called an inner product space or pre-Hilbert space. Remark 9.1.1: The following properties hold in an inner product space. Let X be an inner product space then, (i) for all α; β 2 C; hαx + βy; z) = αhx; zi + βhy; zi 8 x; y; z 2 X. (ii) for all α 2 C hx; αyi =α ¯hx; yi 8 x; y 2 X (iii) for all α; β 2 C 8 x; y; z 2 X. hx; αy + βzi =α ¯hx; yi + β¯hx; zi. 3 Chapter 9 Inner Product Spaces and Hilbert Spaces Remark 9.1.2: Inner product induces a norm. For this we proceed as follows: Proof: Let X be an inner product space. Take x 2 X p Define jjxjj = + hx; xi (9.1.1) Now, jjxjj ≥ 0 as hx; xi ≥ 0 by (IP.4). Also, jjxjj = 0 iff x = θ by (IP.4) Take α 2 C so, jjαxjj2 = hαx; αxi; α 2 X = αα¯hx; xi; x 2 X = jαj2 jjxjj2; x 2 X so jjαxjj = = jαj jjxjj; x 2 X To prove the triangle inequality we first state and prove Cauchy Schwarz Inequality. Cauchy Schwarz Inequality: jhx; yij ≤ jjxjj jjyjj 8 x; y 2 X (9.1.2) Proof: If y = θX then the result follows trivially. Let y =6 θX . Then for every scalars λ 2 C; hx + λy; x + λyi ≥ 0 =) hx; xi + hx; λyi + hλy; xi + hλy; λyi ≥ 0 (By (IP.3)) =) hx; xi + λ¯hx; yi + λhy; xi + jjλyjj2 ≥ 0 =) jjxjj2 + λ¯hx; yi + λhx; yi + jλj2jjyjj2 ≥ 0 hx; yi Take λ = − hy; yi hx; yihx; yi hx; yihx; yi jhx; yij2 So, jjxjj2 − − + jjyjj2 ≥ 0: jjyjj2 jjyjj2 jjyjj4 jhx; yij2 jhx; yij2 jhx; yij2 So; jjxjj2 − − + ≥ 0 jjyjj2 jjyjj2 jjyjj2 4 Chapter 9 Inner Product Spaces and Hilbert Spaces jhx; yij2 So jjxk2 − ≥ 0 jjyjj2 =) jjxjj2jjyjj2 ≥ jhx; yij2 =) jjxjj jjyjj ≥ jhx; yij So jhx; yij ≤ jjxjj jjyjj Sometimes this inequality is abbreviated as C-S inequality. We see that equality sign will hold if and only if in above derivation hx + λy; x + λyi = 0 =) jjx + λyjj2 = 0 =) x + λy = θX , i.e x and y are linearly dependent. We shall now prove triangle inequality for norm. Now 8 x; y 2 X. Now; jjx + yjj2 = hx + y; x + yi = hx; xi + hx; yi + hy; xi + hy; yi = jjxjj2 + jjyjj2 + hx; yi + hy; xi So; jjx + yjj2 = jjxjj2 + jjyjj2 + hx; yi + hy; xi ≤ jjxjj2 + jjyjj2 + hx; yi + hy; xi ≤ jjxjj2 + jjyjj2 + 2jjxjj jjyjj ( ) 2 = jjxjj + jjyjj So; jjx + yjj ≤ jjxjj + jjyjj: Hence, inner product induces a norm and consequently every inner product space is a normed linear space. Remark 9.1.3: So every inner product space is a metric space and the metric induced by inner product is defined as follows: for all x; y 2 X define d : X × X ! R by p d(x; y) = jjx − yjj = + hx − y; x − yi (9.1.3) Theorem 9.1.1: Every inner product function is a continuous function. (Equivalently, if f : X × X ! C defined by f(x; y) = hx; yi; 8 x; y 2 X then f is continuous). Proof: Let X be an inner product space. Define f : X × X ! C by f(x; y) = hx; yi; 8 x; y 2 X. Now take fxng and fyng be a sequence in X such that xn ! x as 5 Chapter 9 Inner Product Spaces and Hilbert Spaces n ! 1 and yn ! y as n ! 1. So, jjxn − xjj ! 0 as n ! 1 and jjyn − yjj ! 0 as n ! 1. As, xn ! x as n ! 1 then, jjxnjj ! jjxjj as n ! 1. So, fjjxnjjg are bounded. So, there exists a constant M > 0 such that jjxnjj ≤ M; 8 n: Now, jhxn; yni − hx; yij = jhxn; yni − hxn; yi + hxn; yi − hx; yij = jhxn; yn − yi + hxn − x; yij ≤ jhxn; yn − yij + jhxn − x; yi ≤ jjxnjj jjyn − yjj + jjxn − xjj jjyjj [By C-S inequality (9.1.2)] ≤ Mjjyn − yjj + jjxn − xjj jjyjj ! 0 as n ! 1 i.e hxn; yni ! hx; yi as n ! 1, implying that f(xn; yn) ! f(x; y) as n ! 1. So, f is continuous. Theorem 9.1.2 (Parallelogram Law): Let X be an inner product space and let x; y 2 X. Then, ( ) jjx + yjj2 + jjx − yjj2 = 2 jjxjj2 + jjyjj2 Proof: jjx + yjj2 = hx + y; x + yi = hx; xi + hx; yi + hy; xi + hy; yi = jjxjj2 + jjyjj2 + hx; yi + hy; xi (9.1.4) and jjx − yjj2 = hx − y; x − yi = hx; xi + hx; −yi + ⟨−y; xi + ⟨−y; −yi = jjxjj2 + jjyjj2 − hx; yi − hy; xi (9.1.5) Adding (9.1.4) and (9.1.5) we get ( ) jjx + yjj2 + jjx − yjj2 = 2 jjxjj2 + jjyjj2 Theorem 9.1.3 (Polarization Identity): Let X be an inner product space, let x; y 2 X. Then h i 1 hx; yi = jjx + yjj2 − jjx − yjj2 + ijjx − iyjj2 − ijjx − iyjj2 (9.1.6) 4 6 Chapter 9 Inner Product Spaces and Hilbert Spaces Proof: Now; jjx + yjj2 = jjxjj2 + jjyjj2 + hx; yi + hy; xi (9.1.7) jjx − yjj2 = jjxjj2 + jjyjj2 − hx; yi − hy; xi (9.1.8) Replacing y by iy in (9.1.7) and (9.1.8) jjx + iyjj2 = jjxjj2 + jjiyjj2 + hx; iyi + hiy; xi = jjxjj2 + jjyjj2 − ihx; yi + ihy; xi (9.1.9) jjx − iyjj2 = jjxjj2 + jjiyjj2 − hx; iyi − hiy; xi = jjxjj2 + jjyjj2 + ihx; yi − ihy; xi (9.1.10) (9.1.7) − (9.1.8) + i(9.1.9) −i(9.1.10), we get (9.1.6). Hence the result. Theorem 9.1.4: Let X be an inner product space. Then (i) Every Cauchy sequence is bounded (ii) If fxng and fyng are two Cauchy sequences in X then fhxn; ynig is also a Cauchy sequence in C and hence convergence in C. Proof: (i) Let, fxng be a Cauchy sequences in X. Then for " = 1 there exists a positive integer N such that, jjxn − xmjj < 1, whenever, n; m ≥ N. In particular, jjxn − xN jj < 1, whenever n ≥ N. jj jj ≤ jj − jj jj jj jj jj 8 ≥ Now, xn n xn xN + xN < 1 + xN on N. Let, M = max jjx1jj; jjx2jj; ··· ; jjxN−1jj; jjxN jj + 1 so, jjxnjj ≤ M 8 n so, fxng is bounded. (ii) Let fxng, fyng be two Cauchy sequences in X. So, jjxn − xmjj ! 0 as n; m ! 1 and jjyn − ymjj ! 0 as n; m ! 1. Also, by (i) jjxnjj ≤ M for all n and for some M > 0. Similarly jjynjj ≤ K, for some K > 0 and 8 n. Now; jhxn; yni − hxm; ymij = jhxn; yni − hxn; ymi + hxn; ymi − hxm; ymij = jhxn; yn − ymi + hxn − xm; ymij ≤ jhxn; yn − ymij + jhxn − xm; ynij 7 Chapter 9 Inner Product Spaces and Hilbert Spaces ≤ jjxnjj jjyn − ymjj + jjxn − xmjj jjynjj (By C-S inequality) ≤ Mjjyn − ymjj + Kjjxn + xmjj ! 0 as n; m ! 1: So, fhxn; ynig is a Cauchy sequences of scalars in C.
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