INDEX OF TERMINOLOGY 359 real algebraic number, 24, 72, 148, 149, 192 real numbers, 3, 86, 101, 142 real part, 19 rectangle, 181, 184, 194, 203 reduced residue system, 309 refinement, 186 reflexive Banach space, 117 regular Borel measure, 200 regular topological space, 67, 68, 97 Reisz-Fisher theorem, 246 relation, 278 relative field norm, 94 relative topology, 42, 69 remainder term, 148, 163 representation, 233, 265, 267 residue field, 95 resolvent set, 137 Riemann integrable, 209, 212 Riemann integrable function, 186–188, 209 Riemann integral, 186, 188, 209, 212 Riemann’s localization principle, 255 Riemann-Lebesgue lemma, 239, 253 Riesz Representation Theorem, 130 Riesz’s lemma, 139, 140 Riesz-Fisher theorem, 245 right Haar integral, 231 right Haar measure, 227, 228, 230–232 right regular representation, 235, 237 right-hand derivative, 241 ring of integers, 87–89, 95 Rolle’s theorem, 145, 148, 176 root test, 28 rules of arithmetic, 280 saddle point, 169, 170 sawtooth curve, 264 sawtooth function, 265 scalar, 314 scalar multiplication, 314 scalar product, 334 Schr¨ oder-Bernstein theorem, 299 Schwartz space, 248, 255, 261, 267 second category, 80, 81 second derivative test, 168, 169 self-dual, 250 selfadjoint operator, 131, 136, 140 semidirect product, 346 separable metric space, 60, 133, 134 separable topological space, 67, 227 separates points, 82 sequence, 297 sequential covering class, 228 sequentially compact set, 17, 23, 58, 59 set containment, 272 shell, 87, 89, 90 sign of a permutation, 326 simple function, 207, 261 smooth function, 162, 248 smooth hypersurface, 165, 166, 172 space of left cosets, 98, 99 span, 316 spanning set, 316 special linear group, 124, 343, 344 special orthogonal group, 124, 347 special unitary group, 125 Spectral Theorem for Compact Selfadjoint Operators, 136, 140 spectrum, 137, 138 standard basis, 121, 132, 150, 315, 317 step function, 253 Stone’s theorem, 83, 84 Stone-Weierstrass theorem, 78, 82, 85, 104 strictly monotonic decreasing function, 145 strictly monotonic increasing function, 52, 145 strongly continuous representation, 233 strongly convergent sequence, 134 subadditive, 196 subadditive functional, 113–116 subbasis for a topology, 248 subfield, 18, 73, 289 subgroup, 98, 342 subintegral domain, 289 subrepresentation, 233 subring, 289 subset, 272 subspace, 318 subspace topology, 69 superposition, 238 superset, 272 support, 103 supremum, 3, 55, 208, 306 surjective, 292 symmetric bilinear form, 334 symmetric difference, 275 symmetric group, 326 symmetric neighborhood, 9 symplectic group, 124 tangent hyperplane, 165 tangent line, 149, 164 taxicab metric, 34
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