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| Foreword | |
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| Linear Spaces | |
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| Linear Maps | |
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| Algebra of linear maps | |
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| Index of a linear map | |
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| The Hahn-Banach Theorem | |
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| The extension theorem | |
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| Geometric Hahn-Banach theorem | |
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| Extensions of the Hahn-Banach theorem | |
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| Applications of the Hahn-Banach theorem | |
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| Extension of positive linear functionals | |
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| Banach limits | |
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| Finitely additive invariant set functions | |
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| Historical note | |
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| Normed Linear Spaces | |
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| Norms | |
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| Noncompactness of the unit ball | |
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| Isometries | |
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| Hilbert Space | |
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| Scalar product | |
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| Closest point in a closed convex subset | |
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| Linear functionals | |
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| Linear span | |
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| Applications of Hilbert Space Results | |
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| Radon-Nikodym theorem | |
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| Dirichlet's problem | |
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| Duals of Normed Linear Spaces | |
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| Bounded linear functionals | |
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| Extension of bounded linear functionals | |
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| Reflexive spaces | |
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| Support function of a set | |
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| Applications of Duality | |
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| Completeness of weighted powers | |
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| The Muntz approximation theorem | |
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| Runge's theorem | |
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| Dual variational problems in function theory | |
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| Existence of Green's function | |
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| Weak* Convergence | |
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| Uniform boundedness of weakly convergent sequences | |
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| Weak sequential compactness | |
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| Weak convergence | |
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| Applications of Weak Convergence | |
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| Approximation of the [delta] function by continuous functions | |
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| Divergence of Fourier series | |
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| Approximate quadrature | |
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| Weak and strong analyticity of vector-valued functions | |
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| Existence of solutions of partial differential equations | |
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| The representation of analytic functions with positive real part | |
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| The Weak and Weak* Topologies | |
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| Locally Convex Topologies and the Krein-Milman Theorem | |
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| Separation of points by linear functionals | |
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| The Krein-Milman theorem | |
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| The Stone-Weierstrass theorem | |
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| Choquet's theorem | |
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| Examples of Convex Sets and Their Extreme Points | |
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| Positive functionals | |
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| Convex functions | |
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| Completely monotone functions | |
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| Theorems of Caratheodory and Bochner | |
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| A theorem of Krein | |
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| Positive harmonic functions | |
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| The Hamburger moment problem | |
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| G. Birkhoff's conjecture | |
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| De Finetti's theorem | |
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| Measure-preserving mappings | |
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| Historical note | |
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| Bounded Linear Maps | |
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| Boundedness and continuity | |
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| Strong and weak topologies | |
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| Principle of uniform boundedness | |
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| Composition of bounded maps | |
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| The open mapping principle | |
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| Historical note | |
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| Examples of Bounded Linear Maps | |
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| Boundedness of integral operators | |
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| The convexity theorem of Marcel Riesz | |
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| Examples of bounded integral operators | |
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| Solution operators for hyperbolic equations | |
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| Solution operator for the heat equation | |
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| Singular integral operators, pseudodifferential operators and Fourier integral operators | |
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| Banach Algebras and their Elementary Spectral Theory | |
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| Normed algebras | |
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| Functional calculus | |
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| Gelfand's Theory of Commutative Banach Algebras | |
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| Applications of Gelfand's Theory of Commutative Banach Algebras | |
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| The algebra C(S) | |
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| Gelfand compactification | |
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| Absolutely convergent Fourier series | |
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| Analytic functions in the closed unit disk | |
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| Analytic functions in the open unit disk | |
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| Wiener's Tauberian theorem | |
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| Commutative B-algebras | |
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| Historical note | |
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| Examples of Operators and Their Spectra | |
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| Invertible maps | |
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| Shifts | |
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| Volterra integral operators | |
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| The Fourier transform | |
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| Compact Maps | |
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| Basic properties of compact maps | |
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| The spectral theory of compact maps | |
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| Historical note | |
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| Examples of Compact Operators | |
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| Compactness criteria | |
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| Integral operators | |
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| The inverse of elliptic partial differential operators | |
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| Operators defined by parabolic equations | |
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| Almost orthogonal bases | |
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| Positive compact operators | |
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| The spectrum of compact positive operators | |
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| Stochastic integral operators | |
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| Inverse of a second order elliptic operator | |
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| Fredholm's Theory of Integral Equations | |
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| The Fredholm determinant and the Fredholm resolvent | |
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| The multiplicative property of the Fredholm determinant | |
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| The Gelfand-Levitan-Marchenko equation and Dyson's formula | |
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| Invariant Subspaces | |
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| Invariant subspaces of compact maps | |
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| Nested invariant subspaces | |
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| Harmonic Analysis on a Halfline | |
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| The Phragmen-Lindelof principle for harmonic functions | |
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| An abstract Pragmen-Lindelof principle | |
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| Asymptotic expansion | |
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| Index Theory | |
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| The Noether index | |
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| Historical note | |
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| Toeplitz operators | |
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| Hankel operators | |
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| Compact Symmetric Operators in Hilbert Space | |
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| Examples of Compact Symmetric Operators | |
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| Convolution | |
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| The inverse of a differential operator | |
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| The inverse of partial differential operators | |
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| Trace Class and Trace Formula | |
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| Polar decomposition and singular values | |
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| Trace class, trace norm, and trace | |
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| The trace formula | |
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| The determinant | |
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| Examples and counterexamples of trace class operators | |
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| The Poisson summation formula | |
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| How to express the index of an operator as a difference of traces | |
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| The Hilbert-Schmidt class | |
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| Determinant and trace for operator in Banach spaces | |
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| Spectral Theory of Symmetric, Normal, and Unitary Operators | |
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| The spectrum of symmetric operators | |
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| Functional calculus for symmetric operators | |
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| Spectral resolution of symmetric operators | |
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| Absolutely continuous, singular, and point spectra | |
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| The spectral representation of symmetric operators | |
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| Spectral resolution of normal operators | |
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| Spectral resolution of unitary operators | |
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| Historical note | |
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| Spectral Theory of Self-Adjoint Operators | |
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| Spectral resolution | |
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| Spectral resolution using the Cayley transform | |
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| A functional calculus for self-adjoint operators | |
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| Examples of Self-Adjoint Operators | |
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| The extension of unbounded symmetric operators | |
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| Examples of the extension of symmetric operators; deficiency indices | |
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| The Friedrichs extension | |
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| The Rellich perturbation theorem | |
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| The moment problem | |
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| Historical note | |
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| Semigroups of Operators | |
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| Strongly continuous one-parameter semigroups | |
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| The generation of semigroups | |
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| The approximation of semigroups | |
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| Perturbation of semigroups | |
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| The spectral theory of semigroups | |
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| Groups of Unitary Operators | |
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| Stone's theorem | |
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| Ergodic theory | |
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| The Koopman group | |
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| The wave equation | |
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| Translation representation | |
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| The Heisenberg commutation relation | |
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| Historical note | |
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| Examples of Strongly Continuous Semigroups | |
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| Semigroups defined by parabolic equations | |
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| Semigroups defined by elliptic equations | |
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| Exponential decay of semigroups | |
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| The Lax-Phillips semigroup | |
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| The wave equation in the exterior of an obstacle | |
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| Scattering Theory | |
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| Perturbation theory | |
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| The wave operators | |
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| Existence of the wave operators | |
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| The invariance of wave operators | |
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| Potential scattering | |
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| The scattering operator | |
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| The Lax-Phillips scattering theory | |
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| The zeros of the scattering matrix | |
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| The automorphic wave equation | |
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| A Theorem of Beurling | |
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| The Hardy space | |
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| Beurling's theorem | |
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| The Titchmarsh convolution theorem | |
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| Historical note | |
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| Texts | |
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| Riesz-Kakutani representation theorem | |
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| Positive linear functionals | |
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| Volume | |
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| L as a space of functions | |
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| Measurable sets and measure | |
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| The Lebesgue measure and integral | |
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| Theory of distributions | |
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| Definitions and examples | |
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| Operations on distributions | |
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| Local properties of distributions | |
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| Applications to partial differential equations | |
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| The Fourier transform | |
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| Applications of the Fourier transform | |
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| Fourier series | |
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| Zorn's Lemma | |
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| Author Index | |
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| Subject Index | |