This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modem mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modem physics, and partial differenrial equations.

Preface
Introduction
Contents of Other Volumes
I: PRELIMINARIES
　1. Sets and functions
　2. Metric and normed linear spaces
　　Appendix Lira sup and lim inf
　3. The Lebesgue integral
　4. Abstract measure theory
　5. Two conrergence arguments
　6. Equicontinuity
　Notes
　Problems
II: HILBERT SPACES
　1. The geometry of Hilbert space
　2. The Riesz lemma
　3. Orthonormal bases
　4. Tensor products of Hilbert spaces
　5. Ergodic theory: an introduction
　Notes
　Problems
III:　BANACH　SPACES
　1. Definition　and　examples
　2. Duals　and　double　duals
　3. The　Hahn-Banach　theorem
　4. Operations　on　Banach　spaces
　5. The　Baire category　theorem　and　its　consequences
　Notes
　Problems
IV:　TOPOLOGICAL　SPACES
　1. General　notions
　2. Nets　and　Convergence
　3. Compactness
　　Appendix The　Stone-Weierstrass　theorem
　4. Measure　theory　on Compact　spaces
　5. Weak　topologies　on　Banach　spaces
　Appendix Weak　and　strong　measurability
　Notes
　Problems
V: LOCALLY ONVEX　SPACES
　1. General　properties
　2. Frdchet　spaces
　3. Functions　of　rapid　decease　and　the　tempered　distributions
　　Appendix The　N-representation　for　and
　4. Inductive　limits:　generalized　functions　and　weak　solutions　of partial　differential　equations
　5. Fixed　point　theorems
　6. Applications　of　fixed　point　theorems
　7. Topologies　on　locally　convex　spaces:　duality　theory　and　the strong　dual　topology
　　　Appendix Polars　and　the　Mackey-Arens　theorem
　Notes
　Problems
VI: BOUNDED　OPERATORS
VII: THE　SPECTRAL　THEOREM
VIII: UNBOUNDED　OPERATORS
THE　FOURIER　TRANSFORM
SUPPLEMENTARY　MATERIAL
List　of　Symbols
Index