Measure, Integration & Real Analysis, 2021a

Contents
xi
9 Real and Complex Measures 255
9A Total Variation 256
Properties of Real and Complex Measures 256
Total Variation Measure 259
The Banach Space of Measures 262
Exercises 9A 265
9B Decomposition Theorems 267
Hahn Decomposition Theorem 267
Jordan Decomposition Theorem 268
Lebesgue Decomposition Theorem 270
Radon–Nikodym Theorem 272
Dual Space of L p (μ) 275
Exercises 9B 278
10 Linear Maps on Hilbert Spaces 280
10A Adjoints and Invertibility 281
Adjoints of Linear Maps on Hilbert Spaces 281
Null Spaces and Ranges in Terms of Adjoints 285
Invertibility of Operators 286
Exercises 10A 292
10B Spectrum 294
Spectrum of an Operator 294
Self-adjoint Operators 299
Normal Operators 302
Isometries and Unitary Operators 305
Exercises 10B 309
10C Compact Operators 312
The Ideal of Compact Operators 312
Spectrum of Compact Operator and Fredholm Alternative 316
Exercises 10C 323
10D Spectral Theorem for Compact Operators 326
Orthonormal Bases Consisting of Eigenvectors 326
Singular Value Decomposition 332
Exercises 10D 336
Measure, Integration & Real Analysis, by Sheldon Axler