## A Course in Functional AnalysisFunctional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts. |

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### Contents

Hilbert Spaces | xv |

2 Orthogonality | 5 |

3 The Riesz Representation Theorem | 9 |

4 Orthonormal Sets of Vectors and Bases | 12 |

5 Isomorphic Hilbert Spaces and the Fourier Transform for the Circle | 17 |

6 The Direct Sum of Hilbert Spaces | 21 |

Operators on Hilbert Space | 24 |

2 The Adjoint of an Operator | 29 |

3 Compact Operators | 171 |

4 Invariant Subspaces | 176 |

5 Weakly Compact Operators | 181 |

Banach Algebras and Spectral Theory for Operators on a Banach Space | 185 |

2 Ideals and Quotients | 189 |

3 The Spectrum | 193 |

4 The Riesz Functional Calculus | 197 |

5 Dependence of the Spectrum on the Algebra | 203 |

3 Projections and Idempotents Invariant and Reducing Subspaces | 34 |

4 Compact Operators | 39 |

5 The Diagonalization of Compact SelfAdjoint Operators | 44 |

SturmLiouville Systems | 47 |

7 The Spectral Theorem and Functional Calculus for Compact Normal Operators | 52 |

8 Unitary Equivalence for Compact Normal Operators | 58 |

Banach Spaces | 61 |

2 Linear Operators on Normed Spaces | 65 |

3 Finite Dimensional Normed Spaces | 67 |

4 Quotients and Products of Normed Spaces | 68 |

5 Linear Functionals | 71 |

6 The HahnBanach Theorem | 75 |

Banach Limits | 80 |

Runges Theorem | 81 |

Ordered Vector Spaces | 84 |

10 The Dual of a Quotient Space and a Subspace | 86 |

11 Reflexive Spaces | 87 |

12 The Open Mapping and Closed Graph Theorems | 88 |

13 Complemented Subspaces of a Banach Space | 91 |

14 The Principle of Uniform Boundedness | 93 |

Locally Convex Spaces | 97 |

2 Metrizable and Normable Locally Convex Spaces | 103 |

3 Some Geometric Consequences of the HahnBanach Theorem | 106 |

4 Some Examples of the Dual Space of a Locally Convex Space | 112 |

5 Inductive Limits and the Space of Distributions | 114 |

Weak Topologies | 122 |

2 The Dual of a Subspace and a Quotient Space | 126 |

3 Alaoglus Theorem | 128 |

4 Reflexivity Revisited | 129 |

5 Separability and Metrizability | 132 |

The StoneCech Compactification | 135 |

7 The KreinMilman Theorem | 139 |

The StoneWeierstrass Theorem | 143 |

9 The Schauder Fixed Point Theorem | 147 |

10 The RyllNardzewski Fixed Point Theorem | 149 |

Haar Measure on a Compact Group | 152 |

12 The KreinSmulian Theorem | 157 |

13 Weak Compactness | 161 |

Linear Operators on a Banach Space | 164 |

2 The BanachStone Theorem | 169 |

6 The Spectrum of a Linear Operator | 206 |

7 The Spectral Theory of a Compact Operator | 212 |

8 Abelian Banach Algebras | 216 |

9 The Group Algebra of a Locally Compact Abelian Group | 221 |

CAlgebras | 230 |

2 Abelian CAlgebras and the Functional Calculus in CAlgebras | 234 |

3 The Positive Elements in a CAlgebra | 238 |

4 Ideals and Quotients of CAlgebras | 243 |

5 Representations of CAlgebras and the GelfandNaimarkSegal Construction | 246 |

Normal Operators on Hilbert Space | 253 |

2 The Spectral Theorem | 260 |

3 StarCyclic Normal Operators | 266 |

4 Some Applications of the Spectral Theorem | 269 |

5 Topologies on B H | 272 |

6 Commuting Operators | 274 |

7 Abelian von Neumann Algebras | 279 |

The Conclusion of the Saga | 283 |

9 Invariant Subspaces for Normal Operators | 288 |

A Complete Set of Unitary Invariants | 291 |

Unbounded Operators | 301 |

2 Symmetric and SelfAdjoint Operators | 306 |

3 The Cayley Transform | 314 |

4 Unbounded Normal Operators and the Spectral Theorem | 317 |

5 Stones Theorem | 325 |

6 The Fourier Transform and Differentiation | 332 |

7 Moments | 341 |

Fredholm Theory | 345 |

2 Fredholm Operators | 347 |

3 Fredholm Theory | 350 |

4 The Essential Spectrum | 356 |

5 The Components of J F | 360 |

6 A Finer Analysis of the Spectrum | 361 |

Preliminaries | 367 |

2 Topology | 369 |

The Dual of Lp𝜇 | 373 |

The Dual of C₀X | 376 |

Bibliography | 382 |

389 | |

Index | 393 |

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### Common terms and phrases

abelian algebra with identity assume ball Banach algebra Banach space bijection bounded operator C*-algebra Cb(X closure compact normal operator compact operator compact space compact subset complete continuous function continuous seminorm converges convex subset Corollary countable cyclic vector defined Definition denoted dense easy eigenvalues Example Exercise extreme point fact finite dimensional Fredholm function f functional calculus Hence hermitian Hilbert space homomorphism implies inner product invertible isometrically isomorphic Lemma Let f linear manifold linear operator locally compact maximal ideal measure space metric multiplication nonzero normed space Note open subset orthogonal orthonormal polynomial preceding proposition projection proof reader reflexive representation result Riesz scalar-valued spectral measure self-adjoint operator semi-Fredholm semi-Fredholm operator seminorm sequence space and let spectral measure Spectral Theorem statements are equivalent Suppose surjective symmetric operator Theory unique unitary vector space Verify the statements von Neumann algebra weak topology weak-star weakly compact