CHAPTER 5. INNER PRODUCT SPACES AND LINEAR OPERATORS MODULE 1. INNER PRODUCT SPACES INDRANATH SENGUPTA Contents 1. Inner products and Norms 1 2. Cauchy-Schwarz inequality and Triangle inequality 3 3. Metric 4 We assume that the field F is either R or C. Let V be a vector space over F. 1. Inner products and Norms Definition 1. An inner product on V is a map h ; i : V × V ! F, such that the following properties are satisfied for every x; y; z 2 V and c 2 F: (i) hx + y; zi = hx; zi + hy; zi; (ii) hcx; zi = chx; zi; (iii) hx; zi = hz; xi; (iv) hx; xi > 0, if x 6= 0. Remarks. (1) If F = R, then hx; zi = hx; zi = hz; xi. (2) If h ; i is an inner product on V , then r · h ; i is also an inner product for every r > 0. (3) For a fixed z 2 V , the function ' = h−; zi : V ! F defined as '(x) = hx; zi is a linear map over F. If F = R, then it is also linear with respect to the other variable. In other words, if F = R, then h−; −i : V × V ! F is bilinear over F. 1 2 Module 1 n Example 1.1. Given x = (x1; : : : ; xn) and y = (y1; : : : ; yn) in F , we define hx; yi = x1y1 + x2y2 + ··· + xnyn: This is called the usual inner product or the dot product of vectors in Rn or Cn. Example 1.2. Let V = C[0; 1], the vector space of all real-valued R 1 continuous functions on [0; 1]. Define hf; gi = 0 f(t)g(t)dt. Example 1.3. Let V = C[0; 2π]C denote the vector space of all complex- 1 R 2π valued continuous functions on [0; 2π]. Define hf; gi = 2π 0 f(t)g(t)dt. Definition 2. Let A = (aij) be an n × n matrix over F. The adjoint ∗ t of A is defined to be the matrix A = (aij) . This is nothing but the conjugate transpose of A. i 0 −i 1 − i Example 1.4. if A = , then A∗ = . 1 + i 1 0 1 Example 1.5. V = Mn(F) is an inner product space with respect to the inner product hA; Bi = trace(B∗A). We observe the following identity which is useful: n n n n n ∗ X ∗ X X ∗ X X 2 hA; Ai = trace(A A) = (A A)ii = (A )ikAki = jAkij : i=1 i=1 k=1 i=1 k=1 Definition 3. The pair (V; h ; i) is called an inner product space over the field F if h ; i is an inner product on V . We often hide h ; i and call V an inner product space to mean that there is an inner product h ; i defined on V . The vector space V is called a real inner product space if F = R and a complex inner product space if F = C. It is easy to check that h ; i is an inner product in all the examples discussed above. Theorem 1.6. Let V be an inner product space over F. The following statements are true for every x; y; z 2 V and c 2 F. (i) hx; y + zi = hx; yi + hx; zi; Chapter 5. Inner Product Spaces and Linear Operators 3 (ii) hx; cyi = chx; yi; (iii) hx; 0i = h0; xi = 0; (iv) hx; xi = 0 if and only if x = 0; (v) if hx; yi = hx; zi for all x 2 V , then y = z. Proof. Follows from the definition of h ; i. Definition 4. Let V be an inner product space over F. For every x in V , we define the norm of x to be jjxjj = phx; xi. n n Example 1.7. If V = R and x = (x1; : : : ; xn) 2 R , then 1 p 2 2 2 jjxjj = hx; xi = jx1j + ··· + jxnj ; which is the usual length of the vector x or the Euclidean distance of the vector x from the origin (0;:::; 0). The following proposition is easy to verify using the definition of norm. Proposition 1.8. For every x 2 V and c 2 F, (i) jjc · xjj = jcj · jjxjj; (ii) jjxjj = 0 if and only if x = 0 2. Cauchy-Schwarz inequality and Triangle inequality Theorem 2.1. Let V be an inner product space over F. (i) (Cauchy-Schwarz Inequality) jhx; yij ≤ jjxjj · jjyjj, for all x; y 2 V . (ii) (Triangle Inequality) jjx+yjj ≤ jjxjj+jjyjj, for all x; y 2 V . (iii) jjjxjj − jjyjjj ≤ jjx − yjj, for all x; y 2 V , for all x; y 2 V . Proof. (i) If y = 0 then the statement is true. We therefore assume that y 6= 0. Hence, hy; yi 6= 0. We have hx; yi hx; yi hx; yi 0 ≤ jjx − yjj2 = hx − y; x − yi hy; yi hy; yi hy; yi hx; yi hx; yi hx; yi hx; yi = hx; xi − hx; yi − hy; xi + hy; yi hy; yi hy; yi hy; yi hy; yi jhx; yij2 = jjxjj2 − : jjyjj2 4 Module 1 (ii) jjx + yjj2 ≤ hx + y; x + yi = hx; xi + hy; xi + hx; yi + hy; yi = jjxjj2 + 2Rehx; yi + jjyjj2 ≤ jjxjj2 + 2jhx; yij + jjyjj2 ≤ jjxjj2 + 2jjxjj · jjyjj + jjyjj2 = (jjxjj + jjyjj)2: (iii) jjxjj = jj(x − y) + yjj ≤ jjx − yjj + jjyjj by the triangle inequality. Therefore, jjxjj−jjyjj ≤ jjx−yjj. Interchanging x and y in the inequality we get jjyjj−jjxjj ≤ jjy−xjj. The proof follows from these observations. Remark 2.2. It can be proved easily that jhx; yij = jjxjj · jjyjj if and only y = λx for some λ 2 F. We also not that in case of R2, the Cauchy-Schwarz inequality is easy to prove because jhx; yij = j jjxjj jjyjj cos θ j = jjxjj jjyjj j cos θj ≤ jjxjj jjyjj: 3. Metric Definition 5. Let X be a non-empty set. A metric on X is a function d : X × X ! R such that (i) d(x; y) ≥ 0 for every x; y 2 X. (ii) d(x; y) = 0 if and only if x = y. (iii) d(x; y) = d(y; x) for every x; y 2 X. (iv) (Triangle inequality) d(x; z) ≤ d(x; y)+d(y; z), for all x; y; z 2 X. Theorem 3.1. Let V be an inner product space over R. Then, d(x; y) = jjx − yjj, for x; y 2 V defines a metric on V . Chapter 5. Inner Product Spaces and Linear Operators 5 Proof. Properties (i) - (iii) follow from the definition. We prove the triangle inequality for d. d(x; z) = jjx − zjj = jj(x − y) + (y − z)jj ≤ jj(x − y)jj + jj(y − z)jj = d(x; y) + d(y; z): .