Analysis

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780821827833 | 0821827839
Cover: Hardcover
Copyright: 5/1/2001

Rent

(Recommended)

$34.94

Term

Due

Price

Semester

Dec 13

$36.77

Quarter

Aug 16

$34.94

Short Term

Jul 6

$33.09

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping cart.
Buy New

Usually Ships in 3-5 Business Days

$50.24

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.

Preface to the First Edition

xvii

Preface to the Second Edition

xxi

Measure and Integration

(40)

Introduction

(3)

Basic notions of measure theory

(5)

Monotone class theorem

(2)

Uniqueness of measures

(1)

Definition of measurable functions and integrals

(5)

Monotone convergence

(1)

Fatou's lemma

(1)

Dominated convergence

(2)

Missing term in Fatou's lemma

(2)

Product measure

(1)

Commutativity and associativity of product measures

(1)

Fubini's theorem

(1)

Layer cake representation

(2)

Bathtub principle

(1)

Constructing a measure from an outer measure

(2)

Uniform convergence except on small sets

(1)

Simple functions and really simple functions

(2)

Approximation by really simple functions

(2)

Approximation by C∞ functions

(5)

Exercises

(4)

Lp-Spaces

(38)

Definition of Lp-spaces

(3)

Jensen's inequality

(1)

Holder's inequality

(2)

Minkowski's inequality

(2)

Hanner's inequality

(2)

Differentiability of norms

(1)

Completenes of Lp-spaces

(1)

Projection on convex sets

(1)

Continuous linear functionals and weak convergence

(2)

Linear functionals separate

(1)

Lower semicontinuity of norms

(1)

uniform boundedness principle

(2)

Strongly convergent convex combinations

(1)

The dual of Lp (Ω)

(3)

Convolution

(1)

Approximation by C∞-functions

(3)

Separability of Lp (Rn)

(1)

Bounded sequences have weak limits

(1)

Approximation by C∞c-functions

(1)

Convolutions of functions in dual Lp (Rn)-spaces are continuous

(1)

Hilbert-spaces

(8)

Exercises

(4)

Rearrangement Inequalities

(18)

Introduction

(1)

Definition of functions vanishing at infinity

(1)

Rearrangements of sets and functions

(2)

The simplest rearrangement inequality

(1)

Nonexpansivity of rearrangement

(1)

Riesz's rearrangement inequality in one-dimension

(3)

Riesz's rearrangement inequality

(6)

General rearrangement inequality

(1)

Strict rearrangement inequality

(4)

Exercises

(2)

Integral Inequalities

(26)

Introduction

(1)

Young's inequality

(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

Please wait while the item is added to your bag...

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Analysis

Summary

Table of Contents

Supplemental Materials