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MTH 614: Functional Analysis Semester 2, 2019-2020

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MTH 614: Functional Analysis Semester 2, 2019-2020 MTH 614: Functional Analysis Semester 2, 2019-2020 Dr. Prahlad Vaidyanathan Contents I. Normed Linear Spaces3 1. Review of Linear Algebra ........................... 3 2. Definition and Examples............................ 6 3. Bounded Linear Operators........................... 12 4. Completeness.................................. 18 5. Finite Dimensional Spaces........................... 26 6. Quotient Spaces ................................ 31 II. Hilbert Spaces 35 1. Orthogonality.................................. 35 2. Riesz Representation theorem......................... 40 3. Orthonormal Bases............................... 43 4. Isomorphisms of Hilbert spaces........................ 50 5. The Stone-Weierstrass Theorem........................ 53 6. Application: Fourier Series of L2 functions.................. 56 III. The Dual Space 60 1. The Duals of `p and Lp spaces......................... 60 2. The Hahn-Banach Extension theorem .................... 69 3. Separability and Reflexivity.......................... 74 IV. Operators on Banach Spaces 81 1. Baire Category Theorem............................ 81 2. Principle of Uniform Boundedness ...................... 83 3. Open Mapping and Closed Graph Theorems................. 87 4. Application: Fourier Series of L1 functions.................. 92 V. Weak Topologies 96 1. Weak Convergence............................... 96 2. The Hahn-Banach Separation Theorem.................... 99 3. The Weak Topology ..............................104 4. Weak Sequential Compactness.........................109 5. The Weak-∗ Topology .............................113 6. Weak-∗ Compactness..............................116 VI. Operators on Hilbert Spaces 121 1. Adjoint of an Operator.............................121 2 2. The Spectral Theorem: The Finite Dimensional Case . 127 3. Compact Operators ..............................129 4. The Spectral Theorem: The Compact Self-Adjoint Case . 137 3 I. Normed Linear Spaces 1. Review of Linear Algebra All vector spaces in this course will be over k = R or C. Definition 1.1. A Hamel basis for a vector space E over k is a set B ⊂ E such that every element x 2 E can be expressed uniquely as a (finite) linear combination of elements in B. Theorem 1.2. For a subset B ⊂ E, the following are equivalent : (i) B is a Hamel basis for E (ii) B is a maximal linearly independent set (iii) B is a minimal spanning set. (without proof) Theorem 1.3 (Zorn's Lemma). Let (F; ≤) be a partially ordered set such that every totally ordered subset has an upper bound. Then F has a maximal element. (without proof) Theorem 1.4. Every vector space has a basis. In fact, if A ⊂ E is any linearly inde- pendent set, then 9 a Hamel basis B of E such that A ⊂ B. Example 1.5. n (i) Standard basis fei : 1 ≤ i ≤ ng for E = R (ii) Define c00 = f(xn): xi 2 k; 9N 2 N such that xi = 0 8i ≥ Ng Note that it is a vector space over k (Check!). Write ei as above, and note that fei : i 2 Ng is a basis for c00. (iii) Define c0 = f(xn): xi 2 k; lim xi = 0g i!1 Note that fei : i 2 Ng as above is a linearly independent set, but not a basis for c0. (We will prove later that any basis of c0 must be uncountable) 4 (iv) Let a; b 2 R with a < b, then define C[a; b] := ff :[a; b] ! C continuousg n is a vector space. For n 2 N, let en(x) := x , then fen : n 2 Ng is a linearly independent set, but it is not a basis for C[a; b] (HW) Theorem 1.6. If E is a vector space and B1 and B2 are two bases for E, then jB1j = jB2j. This common number is called the dimension of E. (without proof) Definition 1.7. Let E and F be two vector spaces. (i) A function T : E ! F is said to be a linear transformation or operator if T (αa + b) = αT (a) + T (b) for all a; b 2 E and α 2 k. (ii) We write L(E; F ) for the set of all linear operators from E to F . Note that L(E; F ) is a k-vector space. (iii) If F = k, then a linear transformation T : E ! k is called a linear functional. Example 1.8. (i) Let E = kn;F = km, then any m × n matrix A defines a linear transformation TA : E ! F given by x 7! A(x). Conversely, if T 2 L(E; F ), then the matrix whose columns are fT (ei) : 1 ≤ i ≤ ng defines an m × n matrix A such that T = TA. Hence, there is an isomorphism of vector spaces ∼ L(E; F ) = Mmn(k) given by TA 7! A Note: Given another basis B of E, we get another isomorphism from L(E; F ) ! Mmn(k). Thus, the isomorphism is not canonical (it depends on the choice of basis) (ii) Let E = c00 and define X ' : E ! k given by (xn) 7! xn n2N Note that ' is well-defined and linear. Thus, ' 2 L(c00; k) (iii) Let E = C[a; b] and define Z 1 ' : E ! k given by '(f) := f(t)dt 0 Note that ' is linear and so ' 2 L(C[a; b]; k) 5 (iv) Let E = F = C[0; 1]. Define Z x T : E ! F given by T (f)(x) := f(t)dt 0 Note that T is well-defined (from Calculus) and linear. Thus, T 2 L(E; F ). Definition 1.9. (i) Let E; F be two k-vector spaces, then E × F is a vector space under the usual component-wise operations. This is called the external direct sum and is denoted by E ⊕ F . (ii) Let E be a vector space and F1;F2 ⊂ E be two subspaces such that E = F1 + F2 and F1 \ F2 = f0g, then there is an isomorphism ∼ F1 ⊕ F2 = E Thus, E is called the internal direct sum of F1 and F2 and we once again write E = F1 ⊕ F2. Definition 1.10. Let E be a vector space F ⊂ E a subspace (i) The quotient space, denoted by E=F , is the quotient group, viewing E as an abelian group under addition, and F as a (normal) subgroup. Note that E=F has a natural vector space structure given by α(x + F ) := αx + F 8α 2 k; x 2 E (ii) Note that natural the quotient map π : E ! E=F given by x 7! x + F is a surjective linear transformation such that ker(π) = F . (iii) Furthermore, we define the codimension of F by codim(F ) := dim(E=F ) (iv) If codim(F ) = 1, then we say that F is a hyperplane of E. Theorem 1.11. Let E be a finite dimensional vector space and F < E. Then codim(F ) = dim(E) − dim(F ) (HW) Theorem 1.12 (First Isomorphism Theorem). Let T : E ! F be a linear transforma- tion of k-vector spaces. Then (i) ker(T ) < E and Image(T ) < F (ii) Furthermore, E= ker(T ) ∼= Image(T ) as vector spaces. Theorem 1.13 (Rank-Nullity theorem). If T : E ! F is a linear transformation and E is finite dimensional, then dim(ker(T )) + dim(Image(T )) = dim(E) Proof. Theorem 1.11+ Theorem 1.12. (End of Day 1) 6 2. Definition and Examples Definition 2.1. A norm on a k-vector space E is a function k · k : E ! R+ which satisfies the following for all x 2 E; α 2 k (i) kxk ≥ 0 and kxk = 0 iff x = 0 (ii) kαxk = jαjkxk (iii) (Triangle inequality) kx + yk ≤ kxk + kyk The pair (E; k · k) is called a normed linear space. Remark 2.2. If (E; k · k) is a normed linear space, then (i) The function d(x; y) := kx − yk defines a metric on E. This is called the metric induced by the norm. This makes E a Hausdorff topological space. (ii) Note that a sequence (xn) 2 E converges to a point x 2 E iff limn!1 kxn − xk = 0 (iii) By the triangle inequality, vector space addition is a continuous map from E×E ! E (iv) Similarly, scalar multiplication is also a continuous map from k × E ! E (v) By the triangle inequality, it follows that jkxk − kykj ≤ kx − yk and thus the norm function E ! R+ is continuous. Example 2.3. (i) k with absolute value norm (ii) kn with Pn (i) 1-norm given by k(x1; x2; : : : ; xn)k1 := i=1 jxij (ii) sup norm given by k(x1; x2; : : : ; xn)k1 := sup1≤i≤n jxij (iii) c00 with 1-norm or sup norm (iv) c0 with sup norm (v) C[a; b] with R b (i) 1-norm given by kfk1 := a jf(t)jdt. Note that it is a norm because if kfk1 = 0 and f is continuous, then f ≡ 0. (ii) sup-norm given by kfk1 := supx2[a;b] jf(x)j 7 Definition 2.4. An inner product on a vector space E is a function h·; ·i : E × E ! k such that for all x; y; z 2 E; α; β 2 k, we have (i) hαx + βy; zi = αhx; zi + βhy; zi (ii) hx; yi = hy; xi (iii) hx; xi ≥ 0 and hx; xi = 0 iff x = 0 Lemma 2.5 (Cauchy-Schwartz inequality). If E is an inner product space and x; y 2 E, then jhx; yij2 ≤ hx; xihy; yi Proof. The inequality is clearly true if x = 0, so if x 6= 0, set hy; xi z := y − x hx; xi Then hz; xi = 0, so 0 ≤ hz; zi = hz; yi hy; xi = y − x; y hx; xi hy; xihx; yi = hy; yi − hx; xi jhx; yij2 = hy; yi − hx; xi Corollary 2.6. If E is an inner product space, then the function kxk := phx; xi defines a norm on E. This is called the norm induced by the inner product. Proof. We only check the triangle inequality, since the other axioms are obvious.
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