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Functional Analysis Lecture Notes Chapter 1. Hilbert Spaces

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FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 1. HILBERT SPACES CHRISTOPHER HEIL 1. Elementary Properties of Hilbert Spaces Notation 1.1. Throughout, F will denote either the real line R or the complex plane C. All vector spaces are assumed to be over the field F. Definition 1.2 (Semi-Inner Product, Inner Product). If X is a vector space over the field F, then a semi-inner product on X is a function h·; ·i: X × X ! F such that (a) hx; xi ≥ 0 for all x 2 X, (b) hx; yi = hy; xi for all x, y 2 X, and (b) Linearity in the first variable: hαx + βy; zi = αhx; zi + βhy; zi for all x, y, z 2 X and α, β 2 F. Exercise 1.3. Immediate consequences are: (a) Anti-linearity in the second variable: hx; αy + βzi = α¯hx; yi + β¯hx; zi. (b) hx; 0i = 0 = h0; yi. (c) h0; 0i = 0. Definition 1.4. If a semi-inner product h·; ·i satisfies: hx; xi = 0 =) x = 0; then it is called an inner product, and X is called an inner product space or a pre-Hilbert space. Notation 1.5. There are many different standard notations for a semi-inner product, e.g., hx; yi = [x; y] = (x; y) = hxjyi; to name a few. We shall prefer the notation hx; yi. Date: February 20, 2006. These notes closely follow and expand on the text by John B. Conway, \A Course in Functional Analysis," Second Edition, Springer, 1990. 1 2 CHRISTOPHER HEIL Exercise 1.6. Prove the following. n (a) The dot product x · y = x1y¯1 + · · · + xny¯n is an inner product on C . (b) If w1; : : : ; wn ≥ 0 are fixed scalars, then the weighted dot product hx; yi = x1y¯1w1 + n · · · + xny¯nwn is a semi-inner product on C . It is an inner product if wi > 0 for each i. (c) If A is an n × n positive semi-definite matrix (Ax · x ≥ 0 for all x 2 Cn), then hx; yi = Ax · y is a semi-inner product on Cn, and it is an inner product if A is positive definite (meaning Ax · x > 0 for all x =6 0). The weighted dot product is just the special case where A is diagonal. (d) Show that if h·; ·i is an arbitrary inner product on Cn, then there exists a positive definite matrix A such that hx; yi = Ax · y. Hint: Let e1; : : : ; en be the standard basis. Then hx; yi = y¯1hx; e1i + · · · + y¯nhx; eni. For each k, the mapping x 7! hx; eki is linear. Exercise 1.7. Let I be a countable index set (e.g., I = N or Z). Let w : I ! [0; 1). Let 2 2 2 `w = `w(I) be the weighted ` space defined by 2 2 2 `w = `w(I) = x = (xi)i2I : jxij w(i) < 1 : n Xi2I o Show that hx; yi = xi y¯i w(i) i2I X 2 defines a semi-inner product on `w. If w(i) > 0 for all i then it is an inner product. If w(i) = 1 for all i, then we simply call this space `2. The series defining hx; yi converges because of the Cauchy{Schwarz inequality. Example 1.8. Let (X; Ω; µ) be a measure space (X is a set, Ω is a σ-algebra of subsets of of X, and µ: Ω ! [0; 1] is a countably additive measure). Define L2(X) = f : X ! F : jf(x)j2 dµ(x) < 1 ; X n Z o where we identify functions that are equal almost everywhere, i.e., f = g a.e. () µfx 2 X : f(x) =6 g(x)g = 0: Then hf; gi = f(x) g(x) dµ(x) ZX defines an inner product on L2(X). Other notations for L2(X) are L2(µ), L2(X; µ), L2(dµ), L2(X; dµ), etc. 2 2 The space `w(I) is a special case of L (X), where X = I and µ is a weighted counting measure on I. CHAPTER 1. HILBERT SPACES 3 Exercise 1.9. Every subspace of an inner product space is itself an inner product space (using the same inner product). 2 2 Hence every subspace of `w or L (X) is also an inner product space. For example, V = L2(Rn) \ C(Rn) = ff 2 L2(Rn) : f is continuousg is a subspace of L2(Rn) and hence is also an inner product space. List some other subspaces 2 2 of some particular `w or L (X) spaces. Notation 1.10 (Associated Semi-Norm or Norm). If h·; ·i is a semi-inner product on X, then we will write kxk = hx; xi1=2: We will see later that k · k defines a semi-norm on X, and that it is a norm on X if h·; ·i is an inner product on X. Lemma 1.11 (Polar Identity). If h·; ·i is a semi-inner product on X, then for all x, y 2 X we have kx + yk2 = kxk2 + 2 Re hx; yi + kyk2: Proof. Using the fact that z + z¯ = 2Re z, we have kx + yk2 = hx + y; x + yi = kxk2 + hx; yi + hy; xi + kyk2 = kxk2 + 2 Re hx; yi + kyk2: Theorem 1.12 (Cauchy-Bunyakowski-Schwarz Inequality). If h·; ·i is a semi-inner product on X, then 8 x; y 2 X; jhx; yij ≤ kxk kyk: Moreover, equality holds if and only if there exist scalars α, β 2 F, not both zero, such that kαx + βyk = 0. Proof. If x, y 2 X and α 2 F, then 0 ≤ kx − αyk2 = hx − αy; x − αyi = hx; xi − αhy; xi − α¯hx; yi + αα¯hy; yi = kxk2 − αhy; xi − α¯hx; yi + jαj2 kyk2: Write hy; xi = beiθ where b ≥ 0 and θ 2 R (θ = 0 if F = R). Set α = e−iθt where t 2 R. Then for this α we have 0 ≤ kxk2 − e−iθtbeiθ − eiθtbe−iθ + t2 kyk2 = kxk2 − 2bt + kyk2 t2: This is a quadratic polynomial in t. In order for this polynomial to be everywhere nonneg- ative, it can have at most one real root, which means that the discriminant must be ≤ 0, i.e., (−2b)2 − 4 kyk2 kxk2 ≤ 0: 4 CHRISTOPHER HEIL Hence jhx; yij2 = jhy; xij2 = b2 ≤ kxk2 kyk2: Exercise: Supply the proof for the case of equality. Corollary 1.13. If h·; ·i is a semi-inner product on X, then kxk = hx; xi1=2 defines a semi- norm on X, which means that: (a) kxk ≥ 0 for all x 2 X, (b) kαxk = jαj kxk for all x 2 X and α 2 F, (c) Triangle Inequality: kx + yk ≤ kxk + kyk for all x, y 2 X. If h·; ·i is an inner product on X then k · k is a norm on X, which means that in addition to (a){(c) above, we also have: (d) kxk = 0 =) x = 0. Proof. (a), (b), (d) Exercises. (c) Using the Polar Identity, we have kx + yk2 = hx + y; x + yi = kxk2 + 2 Re hx; yi + kyk2 ≤ kxk2 + 2 jhx; yij + kyk2 ≤ kxk2 + 2 kxk kyk + kyk2 2 = kxk + kyk : Definition 1.14 (Distance). Let h·; ·i be an inner product on X. Then the distance from x to y in X is d(x; y) = kx − yk: Exercise 1.15. d(·; ·) defines a metric on X. Definition 1.16. Many of the results in this chapter are valid not only for inner product spaces, but for any space which possesses a norm. Assume that X is a vector space. A semi-norm on X is a function k · k: X ! [0; 1) such that statements (a)-(c) above hold. If, in addition, statement (d) holds, then k · k is a norm, and X is called a normed space, normed linear space, or normed vector space. Definition 1.17 (Convergence). Let X be a normed linear space (such as an inner product 1 space), and let ffngn=1 be a sequence of elements of X. CHAPTER 1. HILBERT SPACES 5 (a) We say that ffng converges to f 2 X, and write fn ! f, if lim kf − fnk = 0; n!1 i.e., 8 " > 0; 9 N > 0 such that n > N =) kf − fnk < ": (b) We say that ffng is Cauchy if 8 " > 0; 9 N > 0 such that m; n > N =) kfm − fnk < ": Exercise 1.18. Let X be a normed linear space. Prove the following. (a) Reverse Triangle Inequality: kxk − kyk ≤ kx − yk. (b) Continuity of the norm: xn ! x =) k xnk ! kxk. (c) Continuity of the inner product: If X is an inner product space then xn ! x; yn ! y =) hxn; yni ! hx; yi: (d) All convergent sequences are bounded, and the limit of a convergent sequence is unique. (e) Cauchy sequences are bounded. (f) Every convergent sequence is Cauchy. (g) There exist inner product spaces for which not every Cauchy sequence is convergent. Definition 1.19 (Hilbert Space). An inner product space H is called a Hilbert space if it is complete, i.e., if every Cauchy sequence is convergent. That is, 1 ffngn=1 is Cauchy in H =) 9 f 2 H such that fn ! f: The letter H will always denote a Hilbert space. Definition 1.20 (Banach Space). A normed linear space X is called a Banach space if it is complete, i.e., if every Cauchy sequence is convergent. We make no assumptions about the meaning of the symbol X, i.e., it need not denote a Banach space. A Hilbert space is thus a Banach space whose norm is associated with an inner product.
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