Measure, Integration & Real Analysis, 2021a

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Contents
6E Consequences of Baire’s Theorem 184
7 L p Spaces 193
Baire’s Theorem 184
Open Mapping Theorem and Inverse Mapping Theorem 186
Closed Graph Theorem 188
Principle of Uniform Boundedness 189
Exercises 6E 190
7A L p (μ) 194
Hölder’s Inequality 194
Minkowski’s Inequality 198
Exercises 7A 199
7B L p (μ) 202
8 Hilbert Spaces 211
Definition of L p (μ) 202
L p (μ) Is a Banach Space 204
Duality 206
Exercises 7B 208
8A Inner Product Spaces 212
Inner Products 212
Cauchy–Schwarz Inequality and Triangle Inequality 214
Exercises 8A 221
8B Orthogonality 224
Orthogonal Projections 224
Orthogonal Complements 229
Riesz Representation Theorem 233
Exercises 8B 234
8C Orthonormal Bases 237
Bessel’s Inequality 237
Parseval’s Identity 243
Gram–Schmidt Process and Existence of Orthonormal Bases 245
Riesz Representation Theorem, Revisited 250
Exercises 8C 251
Measure, Integration & Real Analysis, by Sheldon Axler