INDEX Banach ∗-algebra, 189 Almost everywhere p Banach limits, 38 (a.e.p), 167 Absolute value of an Banach means, 38 operator, 216 Banach space, 65 Adjoining an identity, 191 characterization, 66 Adjoint, 93 complex, 65 of an operator, 157 reflexive, 93 Alaoglu’s Theorem, 94 reflexive criterion, 96 Algebraic direct sum, 2 Bessel’s Inequality, 147 Approximate identities, 104 Basis: for a vector space, 2 Baire measures, 26 for a topology, 5 Baire sets, 26 orthonormal, 150 Banach algebra, 75, 166, 175 Bilateral shift, 159 adjoining an identity, 191 adjoint, 189 Borel function, 19, 165 Gelfand transform, 195 Borel measure, 19 L1(R), 190 Borel sets, 19, 165 normal element, 189 Borel space, 167 projection element, 189 selfadjoint subset, 189 ∗ structure space, 194 C -algebra, 190 ∗ structure space sub C -algebra, 190 of L1(R), 196 Cartesian products, 1 unitary element, 189 Cauchy-Schwarz Inequality, 142 Cayley transform, 159 Convolution kernels, 103, 104 of an unbounded selfadjoint on the circle, 103 operator, 234 operator, 103, 160 Chain rule, 250 Convolution Theorem, 113 Character, 137, 240 Countably normed space, 52 Cyclic vector, 171 Choquet’s Theorem, 61 and the Riesz Represen- tation Theorem, 97 Derivative, 245 Diagonalizable, 206 Closed Graph Theorem, 71 Diagonalizing a Hermitian Closure of an operator, 238 matrix, 207 Cluster point, 35 Diffeomorphism, 246 Cofinal, 35 Differentiability, 245 Compact operator, 221 Differentiable manifold, 261 characterization, 223 Differential, 247 properties of the set, 223 Dimension (vector space), 2 Compact space, 5 Direct product, 3 locally compact, 5 Direct sum, 3 σ-compact, 5, 20 Hilbert spaces, 153 Compact support, 19 normed linear spaces, 65 Completion of a normed projection-valued linear space, 75 measures, 169 Complex Banach space, 65 Directed set, 35 Discontinuous linear Complex measure, 24, 91 norm, 92 functionals, 50 Discrete spectrum, 217 Cone, 28 Conjugate linear, 143 Distribution, 86 Conjugate space, 90 properties, 87 tempered function, 86 Continuity (characterized), 49 tempered measure, 87 Continuity and the graph, 50 Distributional derivative, 87 Continuous spectrum, 217 Distributions as derivatives Continuously differen- of functions, 88 tiable, 253 Domain of an unbounded Contraction map, 255 selfadjoint operator, 233 Contraction Mapping Dual space, 81 Theorem, 256 Duality Theorem, 83 Convex hull, 54 Eigenvalue, 158 Convex set, 54 Eigenvector, 158 Essential spectrum, 217 Hahn-Banach Theorem: Essentially selfadjoint, 238 complex version, 41 extreme point version, 59 Extreme point, 59 locally convex version, 56 norm version, 33, 75 Face, 59 positive cone version, 28 Finite intersection semigroup-invariant version, 39 property, 8 seminorm version, 32 Finite rank operator, 221 Hausdorff space, 5 Finitely additive measure, 40 Hausdorff-Young translation-invariant, 41 Inequality, 117 Hausdorff Maximality First derivative test, 251 Principle, 1 Fourier transform, 112 Hermitian form, 143, 156 inversion Theorem, 114 Hilbert space direct sum, 153 Gauss kernel, 113 Hilbert transform, 120 on n, 118 R Hilbert-Schmidt operator, 222 L2 transform, 116 characterization, 225 Plancherel Theorem, 114 Hilbert-Schmidt inner tempered distribution, 117 product, 228 Frechet derivative, 247 Hilbert-Schmidt norm, 228 Frechet space, 44, 86 Idempotent operator, 158 Gauss kernel, 108 Implicit Function Fourier transform, 113 Theorem, 256 in Hilbert space, 261 Gelfand transform, 195 foliated version, 260 General spectral Independence of spectrum, 201 Theorem, 202 Infinite Cartesian products, 1 Generator of a one-parameter Inner product, 143 group, 239 Inner product space, 143 Gleason’s Theorem, 240 Integral kernel, 99 Gradient, 262 Integral operator, 99 bounded, 99 Gram-Schmidt process, 147 Invariance of the essential Green’s functions, 108 spectrum, 236 for the Laplacian, 119? Inverse Function Theorem, 259 Multiplicity, 217 Involution, 189 Multipliers on the line, 120 Isometric isomorphism, 65 Isomorphism Theorem, 70 Natural map, 4 Neighborhood, 5 Jacobian, 249 Net, 35 Non-locally convex space, 58 Kernel and range, 3 Norm: Kernel, 99 of a linear functional, 90 convolution, 103 of an operator, 72 Gauss, 108 Normable spaces, 66 Poisson in Rn, 107 Normal operator, 158 Poisson on the circle, 107 Normal topological space, 5 Poisson on the line, 107 reproducing, 104 Observable, 127 singular, 104 compatible, 130 Krein-Milman Theorem, 60 simultaneously, 130 Open Mapping Theorem, 71 Lagrange multipliers, 262 Orthogonal complement, 146 Linear functional, 4 Orthogonal projection, 158 bounded 21 Orthogonality, 146 conjugate linear, 143 Orthonormal basis, 150 norm, 90 Orthonormal set, 146 positive, 12, 28 Linear isomorphism, 3 Parallelogram Law, 146 Linear transformation, 3 Parseval’s Equality, 148 Locally compact, 5 Partial isometry, 215 Locally convex, 54 Plancherel Theorem, 114 Point spectrum, 217 Matrix coefficient, 155 Max norm, 65 Poisson kernel: Mazur’s Theorem, 193 on the circle, 107 on the line, 107 Mean Value Theorem, 252 n Mercer’s Theorem, 233 in R , 107 Metric, 6 Polar decomposition, 215 Metrizable space, 6 Polarization Identity, 116, 145 Minkowski functional, 56 for an operator, 155 Mixed partials, 254 Positive part of a self- Mixed state, 129 adjoint operator, 214 Multiplication operator, 162 Positive cone, 28 Positive functional, 12, 28 Reproducing kernels, 104 Positive operator, 158 Resolution of the Positive square root of a identity, 205 positive operator, 213 Resolvent, 197, 235 Product topology, 7 Restriction of a projection- Projection onto a valued measure, 167 subspace, 152 Riemann-Lebesgue Projection Theorem, 151 Theorem, 113 Projection-valued Riesz Representation measure, 167 Theorem: canonical, 170 for C0(∆), 21 change of variables, 178 complex version, 25 direct sum, 169 in Hilbert space, 153 integral of a bounded for projection- function, 177 valued measures, 179 integral of an unbounded Riesz Interpolation function, 186 Theorem, 76 invariant subspace, 167 Ring of sets, 40 L∞(p), 168 Riesz representation Schwartz space S, 53 Theorem, 179 Second countable, 5 unitarily equivalent, 168 Second derivative test, 255 Pure state, 129 Second dual, 93 Purely atomic spectrum, 217 Self dual, 154 Selfadjoint extension, 239 Questions, 128 complementary, 130 Selfadjoint operator, 158 ordering, 129 unbounded, 233 orthogonality, 131 Selfadjoint subset, 189 sum, 130 Seminorm, 31 Question-valued measure, 132 on a complex space, 41 Quotient space, 4 Separating vector, 171 Quotient topological vector Separation Theorem, 58 space, 48 Sgn (signum function), 78, 117 Quotient topology, 8 Simultaneously diagona- lizable, 207 Reflexive Banach space, 93 Singular kernels, 104 criterion, 96 Spectral mapping Regular topological space, 5 Theorem, 197, 205 Spectral measure, 204 Tangent vector, 261 unbounded selfadjoint Tempered distribution, 86 operator, 234 Tempered function, 86 Spectral radius formula, 199 Tempered measure, 87 Spectral radius, 197 Time evolution, 138 Topologically isomorphic, 44 Spectral Theorem: general,202 Topology on a set, 5 normal operator,202 basis for, 5 selfadjoint operator, 205 product, 7 unbounded selfadjoint relation among the weak, operator, 233 weak∗, and norm, 93 σ-compact, 5, 20 Spectrum, 197, 235 weak and strong, 81 characterization of point relative, 5 spectrum, 217 strong, 81 continuous, 217 weak topologies and discrete, 217 metrizability, 93 essential, 217 weak, 7, 81 independence, 201 weak∗, 83 invariance of essential Total variation norm, 92 spectrum, 236 multiplicity, 217 Trace class operator, 224 point, 217 the set, 225, 228 properties, 220 Trace class norm, 231 purely atomic, 217 Trace of an operator, 231 States, 127 Transpose, 93 mixed, 129 Twice differentiable, 253 pure, 129 Tychonoff Theorem, 36 Stone’s Axiom, 11 Stone’s Theorem, 207, 239 Urysohn’s Lemma, 8 Strong topology, 81 Unbounded selfadjoint operator, 233 Structure space, 194 Cayley transform, 234 of L1( ), 196 R domain, 233 Subadditive functional,31, 55 ∗ positive, 233 Sub C -algebra, 190 resolvent, 235 Subnet, 35 spectral measure, 235 Supporting vector, 171 Spectral Theorem, 233 Supremum norm, 21 spectrum, 235 Symmetry, 139 Uniform Boundedness Principle, 73 Unilateral shift, 159 Unitary operator, 158 characterization, 158 Vanish at infinity, 19 Vector lattice, 11 Vector space direct product, 3 Weak topology, 7, 81 Weak∗ topology, 93 Wigner’s Theorem, 241 Young’s Inequality, 102.