Analysis

by Lieb, Elliott H.; Loss, Michael

• ISBN13: 9780821827833
• ISBN10: 0821827839
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-05-01
• Publisher: Amer Mathematical Society

Summary

Significantly revised and expanded, this new Second Edition provides readers at all levels-from beginning students to practicing analysts-with the basic concepts and standard tools necessary to solve problems of analysis, and how to apply these concepts to research in a variety of areas. Authors Elliott Lieb and Michael Loss take you quickly from basic topics to methods that work successfully in mathematics and its applications. While omitting many usual typical textbook topics, Analysis includes all necessary definitions, proofs, explanations, examples, and exercises to bring the reader to an advanced level of understanding with a minimum of fuss, and, at the same time, doing so in a rigorous and pedagogical way. Many topics that are useful and important, but usually left to advanced monographs, are presented in Analysis, and these give the beginner a sense that the subject is alive and growing.

Table of Contents

Preface to the First Edition	xvii
Preface to the Second Edition	xxi
Measure and Integration	1	(40)
Introduction	1	(3)
Basic notions of measure theory	4	(5)
Monotone class theorem	9	(2)
Uniqueness of measures	11	(1)
Definition of measurable functions and integrals	12	(5)
Monotone convergence	17	(1)
Fatou's lemma	18	(1)
Dominated convergence	19	(2)
Missing term in Fatou's lemma	21	(2)
Product measure	23	(1)
Commutativity and associativity of product measures	24	(1)
Fubini's theorem	25	(1)
Layer cake representation	26	(2)
Bathtub principle	28	(1)
Constructing a measure from an outer measure	29	(2)
Uniform convergence except on small sets	31	(1)
Simple functions and really simple functions	32	(2)
Approximation by really simple functions	34	(2)
Approximation by C∞ functions	36	(5)
Exercises	37	(4)
Lp-Spaces	41	(38)
Definition of Lp-spaces	41	(3)
Jensen's inequality	44	(1)
Holder's inequality	45	(2)
Minkowski's inequality	47	(2)
Hanner's inequality	49	(2)
Differentiability of norms	51	(1)
Completenes of Lp-spaces	52	(1)
Projection on convex sets	53	(1)
Continuous linear functionals and weak convergence	54	(2)
Linear functionals separate	56	(1)
Lower semicontinuity of norms	57	(1)
uniform boundedness principle	58	(2)
Strongly convergent convex combinations	60	(1)
The dual of Lp (Ω)	61	(3)
Convolution	64	(1)
Approximation by C∞-functions	64	(3)
Separability of Lp (Rn)	67	(1)
Bounded sequences have weak limits	68	(1)
Approximation by C∞c-functions	69	(1)
Convolutions of functions in dual Lp (Rn)-spaces are continuous	70	(1)
Hilbert-spaces	71	(8)
Exercises	75	(4)
Rearrangement Inequalities	79	(18)
Introduction	79	(1)
Definition of functions vanishing at infinity	80	(1)
Rearrangements of sets and functions	80	(2)
The simplest rearrangement inequality	82	(1)
Nonexpansivity of rearrangement	83	(1)
Riesz's rearrangement inequality in one-dimension	84	(3)
Riesz's rearrangement inequality	87	(6)
General rearrangement inequality	93	(1)
Strict rearrangement inequality	93	(4)
Exercises	95	(2)
Integral Inequalities	97	(26)
Introduction	97	(1)
Young's inequality	98	(8)
Hardy-Littlewood-Sobolev inequality	106	(4)
Conformal transformations and stereographic projection	110	(4)
Conformal invariance of the Hardy-Littlewood-Sobolev inequality	114	(3)
Competing symmetries	117	(2)
Proof of Theorem 4.3: Sharp version of the Hardy-Littlewood-Sobolev inequality	119	(1)
Action of the conformal group on optimizers	120	(3)
Exercises	121	(2)
The Fourier Transform	123	(12)
Definition of the L1 Fourier transform	123	(2)
Fourier transform of a Gaussian	125	(1)
Plancherel's theorem	126	(1)
Definition of the L2 Fourier transform	127	(1)
Inversion formula	128	(1)
The Fourier transform in Lp (Rn)	128	(1)
The sharp Hausdorff-Young inequality	129	(1)
Convolutions	130	(1)
Fourier transform of /x/αn	130	(1)
Extension of 5.9 to Lp (Rn)	131	(4)
Exercises	133	(2)
Distributions	135	(36)
Introduction	135	(1)
Test functions (The space D (Ω)	136	(1)
Definition of distributions and their convergence	136	(1)
Locally summable functions, Lploc (Ω)	137	(1)
Functions are uniquely determined by distributions	138	(1)
Derivatives of distributions	139	(1)
Definition of W1,p loc (Ω) and W1,p (Ω)	140	(2)
Interchanging convolutions with distributions	142	(1)
Fundamental theorem of calculus for distributions	143	(1)
Equivalence of classical and distributional derivatives	144	(2)
Distributions with zero derivatives are constants	146	(1)
Multiplication and convolution of distributions by C∞-functions	146	(1)
Approximation of distributions by C∞-functions	147	(1)
Linear dependence of distributions	148	(1)
C∞ (Ω) is 'dense' in W1,p loc (Ω)	149	(1)
Chain rule	150	(2)
Derivative of the absolute value	152	(1)
Min and Max of W1,p-functions are in W1,p	153	(1)
Gradients vanish on the inverse of small sets	154	(2)
Distributional Laplacian of Green's functions	156	(1)
Solution of Poisson's equation	157	(2)
Positive distributions are measures	159	(4)
Yukawa potential	163	(3)
The dual of W1,p (Rn)	166	(5)
Exercises	167	(4)
The Sobolev Spaces H1 and H1/2	171	(28)
Introduction	171	(1)
Definition of H1 (Ω)	171	(1)
Completeness of H1 (Ω)	172	(1)
Multiplication by functions in C∞ (Ω)	173	(1)
Remark about H1 (Ω) and W1,2 (Ω)	174	(1)
Density of C∞ (Ω) in H1 (Ω)	174	(1)
Partial integration for functions in H1 (Rn)	175	(2)
Convexity inequality for gradients	177	(2)
Fourier characterization of H1 (Rn)	179	(2)
Heat kernel	180	(1)
Δ is the infinitesimal generator of the heat kernel	181	(1)
Definition of H1/2 (Rn)	181	(3)
Integral formulas for (f/p/f) and (f, √p2 +m2 f)	184	(1)
Convexity inequality for the relativistic kinetic energy	185	(1)
Density of ∞c (Rn) in H1/2 (Rn)	186	(1)
Action of √-Δ and √-Δ + m2 - m on distributions	186	(1)
Multiplication of H1/2 functions by C∞-functions	187	(1)
Symmetric decreasing rearrangement decreases kinetic energy	188	(2)
Weak limits	190	(1)
Magnetic fields: The H1A-spaces	191	(1)
Definition of H1A(Rn)	192	(1)
Diamagnetic inequality	193	(1)
C∞c (Rn) is dense in H1A(Rn)	194	(5)
Exercises	195	(4)
Sobolev Inequalities	199	(38)
Introduction	199	(2)
Definition of D1 (Rn) and D1/2 (Rn)	201	(1)
Sobolev's inequality for gradients	202	(2)
Sobolev's inequality for /p/	204	(1)
Sobolev inequalities in 1 and 2 dimensions	205	(3)
Weak convergence implies strong convergence on small sets	208	(4)
Weak convergence implies a.e. convergence	212	(1)
Sobolev inequalities for Wm,p(Ω)	213	(1)
Rellich-Kondrashov theorem	214	(1)
Nonzero weak convergence after translations	215	(3)
Poincare's inequalities for Wm,p(Ω)	218	(1)
Poincare-Sobolev inequality for Wm,p(Ω)	219	(1)
Nash's inequality	220	(3)
The logarithmic Sobolev inequality	223	(2)
A glance at contraction semigroups	225	(2)
Equivalence of Nash's inequality and smoothing estimates	227	(2)
Application to the heat equation	229	(3)
Derivation of the heat kernel via logarithmic Sobolev inequalities	232	(5)
Exercises	235	(2)
Potential Theory and Coulomb Energies	237	(20)
Introduction	237	(1)
Definition of harmonic, subharmonic, and superharmonic functions	238	(1)
Properties of harmonic, subharmonic, and superharmonic functions	239	(4)
The strong maximum principle	243	(2)
Harnack's inequality	245	(1)
Subharmonic functions are potentials	246	(2)
Spherical charge distributions are 'equivalent' to point charges	248	(2)
Positivity properties of the Coulomb energy	250	(1)
Mean value inequality for Δ -- μ2	251	(3)
Lower bounds on Schrodinger 'wave' functions	254	(1)
Unique solution of Yukawa's equation	255	(2)
Exercises	256	(1)
Regularity of Solutions of Poisson's Equation	257	(10)
Introduction	257	(3)
Continuity and first differentiability of solutions of Poisson's equation	260	(2)
Higher differentiability of solutions of Poisson's equation	262	(5)
Introduction to the Calculus of Variations	267	(32)
Introduction	267	(2)
Schrodinger's equation	269	(1)
Domination of the potential energy by the kinetic energy	270	(4)
Weak continuity of the potential energy	274	(1)
Existence of a minimizer for EO	275	(3)
Higher eigenvalues and eigenfunctions	278	(1)
Regularity of solutions	279	(1)
Uniqueness of minimizers	280	(1)
Uniqueness of positive solutions	281	(1)
The hydrogen atom	282	(2)
The Thomas-Fermi problem	284	(1)
Existence of an unconstrained Thomas-Fermi minimizer	285	(1)
Thomas-Fermi equation	286	(1)
The Thomas-Fermi minimizer	287	(2)
The capacitor problem	289	(4)
Solution of the capacitor problem	293	(3)
Balls have smallest capacity	296	(3)
Exercises	297	(2)
More about Eigenvalues	299	(32)
Min-max principles	300	(2)
Generalized min-max	302	(2)
Bound for eigenvalue sums in a domain	304	(2)
Bound for Schrodinger eigenvalue sums	306	(5)
Kinetic energy with antisymmetry	311	(3)
The semiclassical approximation	314	(2)
Definition of coherent states	316	(1)
Resolution of the identity	317	(2)
Representation of the nonrelativistic kinetic energy	319	(1)
Bounds for the relativistic kinetic energy	319	(1)
Large N eigenvalue sums in a domain	320	(3)
Large N asymptotics of Schrodinger eigenvalue sums	323	(8)
Exercises	327	(4)
List of Symbols	331	(4)
References	335	(6)
Index	341