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Preliminariesp. 1
Linear Algebrap. 2
Metric Spacesp. 11
Lebesgue Integrationp. 20
Normed Spacesp. 31
Examples of Normed Spacesp. 31
Finite-dimensional Normed Spacesp. 39
Banach Spacesp. 45
Inner Product Spaces, Hilbert Spacesp. 51
Inner Productsp. 51
Orthogonalityp. 60
Orthogonal Complementsp. 65
Orthonormal Bases in Infinite Dimensionsp. 72
Fourier Seriesp. 82
Linear Operatorsp. 87
Continuous Linear Transformationsp. 87
The Norm of a Bounded Linear Operatorp. 96
The Space B(X, Y)p. 104
Inverses of Operatorsp. 108
Duality and the Hahn-Banach Theoremp. 121
Dual Spacesp. 121
Sublinear Functionals, Seminorms and the Hahn-Banach Theoremp. 127
The Hahn-Banach Theorem in Normed Spacesp. 132
The General Hahn-Banach theoremp. 137
The Second Dual, Reflexive Spaces and Dual Operatorsp. 144
Projections and Complementary Subspacesp. 155
Weak and Weak-* Convergencep. 159
Linear Operators on Hilbert Spacesp. 167
The Adjoint of an Operatorp. 167
Normal, Self-adjoint and Unitary Operatorsp. 176
The Spectrum of an Operatorp. 183
Positive Operators and Projectionsp. 192
Compact Operatorsp. 205
Compact Operatorsp. 205
Spectral Theory of Compact Operatorsp. 216
Self-adjoint Compact Operatorsp. 226
Integral and Differential Equationsp. 235
Fredholm Integral Equationsp. 235
Volterra Integral Equationsp. 245
Differential Equationsp. 247
Eigenvalue Problems and Green's Functionsp. 253
Solutions to Exercisesp. 265
Further Readingp. 315
Referencesp. 317
Notation Indexp. 319
Indexp. 321
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This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalised to infinite-dimensional spaces.