Functional Analysis F3/F4/NVP (2005) Homework assignment 3
|
|
- Justin Lloyd
- 5 years ago
- Views:
Transcription
1 Functional Analysis F3/F4/NVP (005 Homework assignment 3 All students should solve the following problems: 1. Section 4.8: Problem 8.. Section 4.9: Problem Let T : l l be the operator defined by T x = ( ξ 1 + ξ, ξ, ξ 3 + ξ 4, ξ 4, ξ 5 + ξ 6, ξ 6, ξ 7 + ξ 8,..., x = (ξ j l. Determine the four sets ρ(t, σ p (T, σ c (T, σ r (T. 4. Prove the following (partial converse to the spectral theorem for compact, self-adjoint operators: If H is a Hilbert space, {e n } is a total orthonormal sequence in H, and {λ n } is a sequence of real numbers such that lim n λ n = 0, then there exists a unique bounded linear operator T : H H such that T (e n = λ n e n for all n. This operator is compact and self-adjoint. Students taking Functional Analysis as a 6 point course should also solve the following problems: 5. Section 8.: Problem Let H be the Hilbert space L (, and let T : H H be the multiplication operator T x(t = sin(t x(t. Verify that T is a bounded self-adjoint operator, and find its spectral family. Solutions should be handed in by Thursday, March 3, (Either give the solutions to me directly or put them in my mailbox, third floor, House 3, Polacksbacken. 1
2 Functional Analysis F3/F4/NVP Comments to homework assignment 3 (to be updated! 1.Many students claim that if (x n is a weak Cauchy sequence in X then there must exist some x in X such that (x n converges weakly to x. This is not true (even if X is a Banach space. Example: Let X = c 0 (see p. 70, exercise 1 and let a 1, a, a 3,... be any bounded sequence of complex numbers. Then the following is a weak Cauchy sequence in X = c 0 : x 1 = (a 1, 0, 0, 0,..., x = (a 1, a, 0, 0, 0,..., x 3 = (a 1, a, a 3, 0, 0, 0,..., x 4 = (a 1, a, a 3, a 4, 0, 0, 0,... (Proof: one uses the fact given in exercise 8 on p. 16. In explicit terms this exercise tells us: For every f (c 0 there is some (b n l 1 such that f((ξ n = b nξ n. It follows that if x 1, x, x 3,... is the sequence in c 0 given above then lim j f(x j = b na n. Hence x 1, x, x 3,... is indeed weak Cauchy. However, for most choiches of numbers a 1, a, a 3,..., the above sequence (x n does not converge weakly to an element x c 0.
3 3 Functional Analysis F3/F4/NVP Solutions to homework assignment 3 1. Assume that (x n is a weak Cauchy sequence in X. Take an arbitrary f X. Then (f(x n is a Cauchy sequence in K, and hence it is a bounded sequence. Hence, for every f X there exists some constant c f such that f(x n c f for all n. Now we imitate the argument from p. 58 in the book (part (c: For every x X we define g x X as usual by g x (f = f(x for all f X (cf. p. 39 and Lemma We now have for all n, g xn (f = f(x n c f. This tells us that the sequence (g xn in X = B(X, K is pointwise bounded. We also know that X is a Banach space by Theorem Hence the Uniform Boundedness Theorem (Theorem can be applied, and this gives that there exists a constant c such that g xn c for all n. But x n = g xn by Lemma 4.6-1; hence we have proved that x n c for all n, i.e. the sequence (x n is bounded.. Let X be an arbitrary normed space, and let (f n be a sequence in X. Assume that (f n converges weakly to f X. We wish to prove that (f n is weak* convergent to f. Let x be an arbitrary vector in X. Let g x X be the image of x under the canonical mapping of X into X (cf. p , i.e. g x (h = h(x, h X. Since g x X and (f n converges weakly to f X, we have lim n g x (f n = g x (f. This means lim n f n (x = f(x. Since this holds for all x X, (f n is weak* convergent to f. Now assume that X is reflexive. We wish to prove that weak* convergence in X implies weak convergence in X. Let (f n be an arbitrary sequence in X which is weak* convergent to some f X. Let g be an arbitrary element in X. Then since X is reflexive, there is an element x X such that g x = g. Since (f n is weak* convergent to f we have lim n f n (x = f(x. This can also be written: lim n g x (f n = g x (f. But g x = g, hence we have lim n g(f n = g(f. But g X was arbitrary; hence (f n is weakly convergent to f.
4 4 3. T is a bounded linear operator l l, since, for all (ξ n l, T ((ξ n = ξ 1 + ξ + ξ + ξ 3 + ξ 4 + ξ (1 + 1 ( ξ 1 + ξ + ξ + (1 + 1 ( ξ 3 + ξ 4 + ξ ξ n = 3 (ξ n. (This also shows T 3. In the computation above we used Schwarz inequality on C, which says the following: ( a 1 b 1 + a b ( a 1 + a ( b 1 + b, a 1, a, b 1, b C. (We used this with a 1 = a = 1 and b 1 = ξ 1, b = ξ, etc. To get started, we investigate when T λ = T λi is injective. Let us write x = (ξ n l and y = (T λix = (η n l. Then ( y = (η n = ((1 λξ 1 + ξ, (1 λξ, (1 λξ 3 + ξ 4, (1 λξ 4,... Now if λ = 1 then T x = 0 for every vector x = (ξ n l with ξ = ξ 4 = ξ 6 =... = 0, and there exist many such vectors which are 0, e.g. x = (1, 0, 0, 0,... This shows that if λ = 1 then T λi is not injective; hence 1 σ p (T. From now on we assume λ 1. Then for any x = (ξ n l, formula (** implies: ξ 1 = 1 η 1 1 η ( ξ = 1 η ( ξ 3 = 1 η 3 1 η ( 4 ξ 4 = 1 η 4... Now if some x = (ξ n l would give T x = T x = (η n then each ξ n is given by the same formula as ξ n in (***, i.e. ξ n = ξ n for all n and hence x = x. This proves that T λi is injective (under our present assumption λ 1. From the above computations we also obtain explicit formulas for (T λi 1 and D ((T λi 1 : D ( { } (T λi 1 = (η n l if ξ n is computed using (*** then ξ n < and (T λi ((η 1 n = (ξ n, (η n D ( (T λi 1.
5 To decide whether λ belongs to σ c (T or σ r (T or ρ(t we must determine whether D ((T λi 1 is dense in l and whether (T λi 1 is bounded. In the present problem it turns out to be easiest to consider the first question first. This means trying to bound (ξ n in terms of (η n in (***. For every (η n l we have, if (ξ n is computed using (***: ξ n = 1 η 1 1 η ( + 1 η + 1 η η ( η 4 ( ( η ( 1 + η + 1 η ( ( η ( 3 + η η 4 + Here we used again the inequality (* above. It follows from the above that if we let C λ be the constant C λ = ( + 1, then ξ n C λ η n. In particular, for every (η n l we have ξ n <, i.e. (ξ n l. In view of our earlier formular for D ((T λi 1, this proves that D ( (T λi 1 = l, i.e. D ((T λi 1 is dense in l in the trivial way that it is equal to l. The above computation also proves that (T λi 1 ((η n = ξ n C λ (η n for all (η n l, i.e. the operator (T λi 1 is bounded, with norm C λ. Together, this implies that λ ρ(t. This is true for all λ 1. Hence we have proved: σ p (T = {1}; σ c (T = ; σ r (T = ; ρ(t = C \ {1}. 4. Let {e n } be a total sequence in H, and {λ n } is a sequence of real numbers such that lim n λ n = 0. By the main Theorem 3.5- (in the +...
6 6 version I proved in class, we know that H = { α n e n α n K, α n < } Now if T is a bounded linear operator T : H H with T (e n = λ n e n for all n, we have for all sequences (α n in K with α n < : ( ( N N T α n e n = T α n e n = lim T α n e n N = lim N lim N N α n λ n e n = α n λ n e n. The last sum converges since ( α nλ n sup n λ n α n < (here sup n λ n < since lim n λ n = 0. This proves uniqueness, i.e. that there is only one possible bounded linear operator T : H H with T (e n = λ n e n for all n. On the other hand, the formula T ( α ne n = α nλ n e n truly defines a bounded linear operator T : H H. (Proof: Linearity is checked by a straightforward computation. Furthermore, we have T ( α ne n = α nλ n e n = α nλ n (sup n λ n α n = (sup n λ n α ne n. Hence T is bounded with norm T sup n λ n. We now prove that T is self-adjoint. Let x, y be two arbitrary vectors in H. Then we have x = α ne n and y = β ne n for some α n, β n K with α n <, β n <. Now, by a generalization of the Parseval relation which I have mentioned in class (compare p.175, exercise 4, T x, y = λ n α n e n, β n e n = λ n α n β n and x, T y = α n e n, λ n β n e n = α n λ n β n But all numbers λ n are assumed to be real; hence T x, y = x, T y. Since x, y were arbitrary this proves that T is self-adjoint. Finally, we now prove that T is compact, by imitating the argument in Example 8.1-6, p.409 in the book. For each m Z + we define the
7 operator T m : H H by T m α n e n = m λ n α n e n. This operator is clearly linear, and it is bounded since T m ( m m α n e n = λ n α n sup λ n α n 1 n m ( sup λ n α n e n, 1 n m for all vectors x = α ne n H. (This shows that T m sup1 n m λ n = sup 1 n m λ n. Furthermore the range R(T m Span{e 1, e,..., e m } is finite dimensional, hence by Theorem 8.1-4(a, T m is compact. For each m and each x = (T T m α n e n = hence T T m α ne n H we have ( λ n α n n=m+1 sup λ n n m+1 ( sup λ n α n e n, n m+1 sup n m+1 λ n = sup λ n. n m+1 n=m+1 This number tends to 0 as m, since lim n λ n = 0. Hence T T m 0 as m. Hence by Theorem 8.1-5, the operator T is compact. 7 α n
Functional Analysis F3/F4/NVP (2005) Homework assignment 2
Functional Analysis F3/F4/NVP (25) Homework assignment 2 All students should solve the following problems:. Section 2.7: Problem 8. 2. Let x (t) = t 2 e t/2, x 2 (t) = te t/2 and x 3 (t) = e t/2. Orthonormalize
More informationFunctional Analysis (2006) Homework assignment 2
Functional Analysis (26) Homework assignment 2 All students should solve the following problems: 1. Define T : C[, 1] C[, 1] by (T x)(t) = t x(s) ds. Prove that this is a bounded linear operator, and compute
More informationProve that this gives a bounded linear operator T : X l 1. (6p) Prove that T is a bounded linear operator T : l l and compute (5p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2006-03-17 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE
FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE CHRISTOPHER HEIL 1. Weak and Weak* Convergence of Vectors Definition 1.1. Let X be a normed linear space, and let x n, x X. a. We say that
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More informationFUNCTIONAL ANALYSIS HAHN-BANACH THEOREM. F (m 2 ) + α m 2 + x 0
FUNCTIONAL ANALYSIS HAHN-BANACH THEOREM If M is a linear subspace of a normal linear space X and if F is a bounded linear functional on M then F can be extended to M + [x 0 ] without changing its norm.
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationHomework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator.
Homework 3 1 If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator Solution: Assuming that the inverse of T were defined, then we will have to have that D(T 1
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationMath Solutions to homework 5
Math 75 - Solutions to homework 5 Cédric De Groote November 9, 207 Problem (7. in the book): Let {e n } be a complete orthonormal sequence in a Hilbert space H and let λ n C for n N. Show that there is
More informationMAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
More information285K Homework #1. Sangchul Lee. April 28, 2017
285K Homework #1 Sangchul Lee April 28, 2017 Problem 1. Suppose that X is a Banach space with respect to two norms: 1 and 2. Prove that if there is c (0, such that x 1 c x 2 for each x X, then there is
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationHomework I, Solutions
Homework I, Solutions I: (15 points) Exercise on lower semi-continuity: Let X be a normed space and f : X R be a function. We say that f is lower semi - continuous at x 0 if for every ε > 0 there exists
More informationMath 328 Course Notes
Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationThe weak topology of locally convex spaces and the weak-* topology of their duals
The weak topology of locally convex spaces and the weak-* topology of their duals Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Introduction These notes
More informationExercises to Applied Functional Analysis
Exercises to Applied Functional Analysis Exercises to Lecture 1 Here are some exercises about metric spaces. Some of the solutions can be found in my own additional lecture notes on Blackboard, as the
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)
More informationI teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More information201B, Winter 11, Professor John Hunter Homework 9 Solutions
2B, Winter, Professor John Hunter Homework 9 Solutions. If A : H H is a bounded, self-adjoint linear operator, show that A n = A n for every n N. (You can use the results proved in class.) Proof. Let A
More informationThe Arzelà-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More informationHomework Assignment #5 Due Wednesday, March 3rd.
Homework Assignment #5 Due Wednesday, March 3rd. 1. In this problem, X will be a separable Banach space. Let B be the closed unit ball in X. We want to work out a solution to E 2.5.3 in the text. Work
More informationBiholomorphic functions on dual of Banach Space
Biholomorphic functions on dual of Banach Space Mary Lilian Lourenço University of São Paulo - Brazil Joint work with H. Carrión and P. Galindo Conference on Non Linear Functional Analysis. Workshop on
More informationA Spectral Characterization of Closed Range Operators 1
A Spectral Characterization of Closed Range Operators 1 M.THAMBAN NAIR (IIT Madras) 1 Closed Range Operators Operator equations of the form T x = y, where T : X Y is a linear operator between normed linear
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationReal Variables # 10 : Hilbert Spaces II
randon ehring Real Variables # 0 : Hilbert Spaces II Exercise 20 For any sequence {f n } in H with f n = for all n, there exists f H and a subsequence {f nk } such that for all g H, one has lim (f n k,
More informationIntroduction to Functional Analysis
Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More information3 Orthogonality and Fourier series
3 Orthogonality and Fourier series We now turn to the concept of orthogonality which is a key concept in inner product spaces and Hilbert spaces. We start with some basic definitions. Definition 3.1. Let
More information1 Definition and Basic Properties of Compa Operator
1 Definition and Basic Properties of Compa Operator 1.1 Let X be a infinite dimensional Banach space. Show that if A C(X ), A does not have bounded inverse. Proof. Denote the unit ball of X by B and the
More informationMA5206 Homework 4. Group 4. April 26, ϕ 1 = 1, ϕ n (x) = 1 n 2 ϕ 1(n 2 x). = 1 and h n C 0. For any ξ ( 1 n, 2 n 2 ), n 3, h n (t) ξ t dt
MA526 Homework 4 Group 4 April 26, 26 Qn 6.2 Show that H is not bounded as a map: L L. Deduce from this that H is not bounded as a map L L. Let {ϕ n } be an approximation of the identity s.t. ϕ C, sptϕ
More informationE.7 Alaoglu s Theorem
E.7 Alaoglu s Theorem 359 E.7 Alaoglu s Theorem If X is a normed space, then the closed unit ball in X or X is compact if and only if X is finite-dimensional (Problem A.25). Even so, Alaoglu s Theorem
More informationApplied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.
Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define
More informationProblem 1: Compactness (12 points, 2 points each)
Final exam Selected Solutions APPM 5440 Fall 2014 Applied Analysis Date: Tuesday, Dec. 15 2014, 10:30 AM to 1 PM You may assume all vector spaces are over the real field unless otherwise specified. Your
More information4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan
The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,
More informationThe spectrum of a self-adjoint operator is a compact subset of R
The spectrum of a self-adjoint operator is a compact subset of R Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 Abstract In these notes I prove that the
More informationIntroduction to Bases in Banach Spaces
Introduction to Bases in Banach Spaces Matt Daws June 5, 2005 Abstract We introduce the notion of Schauder bases in Banach spaces, aiming to be able to give a statement of, and make sense of, the Gowers
More informationKernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
More informationLecture Notes in Functional Analysis
Lecture Notes in Functional Analysis Please report any mistakes, misprints or comments to aulikowska@mimuw.edu.pl LECTURE 1 Banach spaces 1.1. Introducton to Banach Spaces Definition 1.1. Let X be a K
More informationSOLUTIONS TO SOME PROBLEMS
23 FUNCTIONAL ANALYSIS Spring 23 SOLUTIONS TO SOME PROBLEMS Warning:These solutions may contain errors!! PREPARED BY SULEYMAN ULUSOY PROBLEM 1. Prove that a necessary and sufficient condition that the
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationFunctional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32
Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak
More informationCOMPACT OPERATORS. 1. Definitions
COMPACT OPERATORS. Definitions S:defi An operator M : X Y, X, Y Banach, is compact if M(B X (0, )) is relatively compact, i.e. it has compact closure. We denote { E:kk (.) K(X, Y ) = M L(X, Y ), M compact
More informationNormed Vector Spaces and Double Duals
Normed Vector Spaces and Double Duals Mathematics 481/525 In this note we look at a number of infinite-dimensional R-vector spaces that arise in analysis, and we consider their dual and double dual spaces
More informationCompact operators on Banach spaces
Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. Appearance of normed spaces. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A - measurable functions on it. 2. For f L 1 (, µ) set f 1
More informationSPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT
SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NIT-Karnataka,
More information************************************* Applied Analysis I - (Advanced PDE I) (Math 940, Fall 2014) Baisheng Yan
************************************* Applied Analysis I - (Advanced PDE I) (Math 94, Fall 214) by Baisheng Yan Department of Mathematics Michigan State University yan@math.msu.edu Contents Chapter 1.
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ.
More informationFredholm Theory. April 25, 2018
Fredholm Theory April 25, 208 Roughly speaking, Fredholm theory consists of the study of operators of the form I + A where A is compact. From this point on, we will also refer to I + A as Fredholm operators.
More informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More informationare Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication
7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =
More informationProjection Theorem 1
Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Wee November 30 Dec 4: Deadline to hand in the homewor: your exercise class on wee December 7 11 Exercises with solutions Recall that every normed space X can be isometrically
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More informationAnalysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
More informationCSE386C METHODS OF APPLIED MATHEMATICS Fall 2014, Final Exam, 9:00-noon, Tue, Dec 16, ACES 6.304
CSE386C METHODS OF APPLIED MATHEMATICS Fall 2014, Final Exam, 9:00-noon, Tue, Dec 16, ACES 6.304 1. Let A be a closed operator defined on a dense subspace D(A) of a Hilbert space U. (a) Define the adjoint
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationRIESZ BASES AND UNCONDITIONAL BASES
In this paper we give a brief introduction to adjoint operators on Hilbert spaces and a characterization of the dual space of a Hilbert space. We then introduce the notion of a Riesz basis and give some
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 22: Preliminaries for Semiparametric Inference
Introduction to Empirical Processes and Semiparametric Inference Lecture 22: Preliminaries for Semiparametric Inference Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationOutline of Fourier Series: Math 201B
Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C
More informationChapter 2 Hilbert Spaces
Chapter 2 Hilbert Spaces Throughoutthis book,h isarealhilbertspacewith scalar(orinner)product. The associated norm is denoted by and the associated distance by d, i.e., ( x H)( y H) x = x x and d(x,y)
More information96 CHAPTER 4. HILBERT SPACES. Spaces of square integrable functions. Take a Cauchy sequence f n in L 2 so that. f n f m 1 (b a) f n f m 2.
96 CHAPTER 4. HILBERT SPACES 4.2 Hilbert Spaces Hilbert Space. An inner product space is called a Hilbert space if it is complete as a normed space. Examples. Spaces of sequences The space l 2 of square
More informationMath 421, Homework #9 Solutions
Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationErrata Applied Analysis
Errata Applied Analysis p. 9: line 2 from the bottom: 2 instead of 2. p. 10: Last sentence should read: The lim sup of a sequence whose terms are bounded from above is finite or, and the lim inf of a sequence
More informationMath 5520 Homework 2 Solutions
Math 552 Homework 2 Solutions March, 26. Consider the function fx) = 2x ) 3 if x, 3x ) 2 if < x 2. Determine for which k there holds f H k, 2). Find D α f for α k. Solution. We show that k = 2. The formulas
More informationWeak Topologies, Reflexivity, Adjoint operators
Chapter 2 Weak Topologies, Reflexivity, Adjoint operators 2.1 Topological vector spaces and locally convex spaces Definition 2.1.1. [Topological Vector Spaces and Locally convex Spaces] Let E be a vector
More information1 Functional Analysis
1 Functional Analysis 1 1.1 Banach spaces Remark 1.1. In classical mechanics, the state of some physical system is characterized as a point x in phase space (generalized position and momentum coordinates).
More informationCHAPTER VIII HILBERT SPACES
CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugate-linear transformation if it is a reallinear transformation from X into Y, and if T (λx)
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationMidterm 1. Every element of the set of functions is continuous
Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions
More informationC.6 Adjoints for Operators on Hilbert Spaces
C.6 Adjoints for Operators on Hilbert Spaces 317 Additional Problems C.11. Let E R be measurable. Given 1 p and a measurable weight function w: E (0, ), the weighted L p space L p s (R) consists of all
More informationFunctional Analysis HW #3
Functional Analysis HW #3 Sangchul Lee October 26, 2015 1 Solutions Exercise 2.1. Let D = { f C([0, 1]) : f C([0, 1])} and define f d = f + f. Show that D is a Banach algebra and that the Gelfand transform
More informationMath 123 Homework Assignment #2 Due Monday, April 21, 2008
Math 123 Homework Assignment #2 Due Monday, April 21, 2008 Part I: 1. Suppose that A is a C -algebra. (a) Suppose that e A satisfies xe = x for all x A. Show that e = e and that e = 1. Conclude that e
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationFunctional Analysis, Stein-Shakarchi Chapter 1
Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More information4 Linear operators and linear functionals
4 Linear operators and linear functionals The next section is devoted to studying linear operators between normed spaces. Definition 4.1. Let V and W be normed spaces over a field F. We say that T : V
More informationFall TMA4145 Linear Methods. Exercise set 10
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 207 Exercise set 0 Please justify your answers! The most important part is how you arrive at
More informationChapter 8 Integral Operators
Chapter 8 Integral Operators In our development of metrics, norms, inner products, and operator theory in Chapters 1 7 we only tangentially considered topics that involved the use of Lebesgue measure,
More informationFRAMES AND TIME-FREQUENCY ANALYSIS
FRAMES AND TIME-FREQUENCY ANALYSIS LECTURE 5: MODULATION SPACES AND APPLICATIONS Christopher Heil Georgia Tech heil@math.gatech.edu http://www.math.gatech.edu/ heil READING For background on Banach spaces,
More informationProblem Set 5: Solutions Math 201A: Fall 2016
Problem Set 5: s Math 21A: Fall 216 Problem 1. Define f : [1, ) [1, ) by f(x) = x + 1/x. Show that f(x) f(y) < x y for all x, y [1, ) with x y, but f has no fixed point. Why doesn t this example contradict
More informationAbout Grupo the Mathematical de Investigación Foundation of Quantum Mechanics
About Grupo the Mathematical de Investigación Foundation of Quantum Mechanics M. Victoria Velasco Collado Departamento de Análisis Matemático Universidad de Granada (Spain) Operator Theory and The Principles
More information2. Dual space is essential for the concept of gradient which, in turn, leads to the variational analysis of Lagrange multipliers.
Chapter 3 Duality in Banach Space Modern optimization theory largely centers around the interplay of a normed vector space and its corresponding dual. The notion of duality is important for the following
More informationElements of Positive Definite Kernel and Reproducing Kernel Hilbert Space
Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space Statistical Inference with Reproducing Kernel Hilbert Space Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department
More informationA note on the σ-algebra of cylinder sets and all that
A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationNOTES ON FRAMES. Damir Bakić University of Zagreb. June 6, 2017
NOTES ON FRAMES Damir Bakić University of Zagreb June 6, 017 Contents 1 Unconditional convergence, Riesz bases, and Bessel sequences 1 1.1 Unconditional convergence of series in Banach spaces...............
More informationFrame expansions in separable Banach spaces
Frame expansions in separable Banach spaces Pete Casazza Ole Christensen Diana T. Stoeva December 9, 2008 Abstract Banach frames are defined by straightforward generalization of (Hilbert space) frames.
More informationMath 699 Reading Course, Spring 2007 Rouben Rostamian Homogenization of Differential Equations May 11, 2007 by Alen Agheksanterian
. Introduction Math 699 Reading Course, Spring 007 Rouben Rostamian Homogenization of ifferential Equations May, 007 by Alen Agheksanterian In this brief note, we will use several results from functional
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More information