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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 | |