360 INDEX OF TERMINOLOGY Taylor expansion, 148 Taylor polynomial, 163, 164, 169, 180 Taylor’s theorem, 147, 148, 163 terminating decimal expansion, 30 Toeplitz, Otto, 141 topological basis, 104, 132 topological group, 96, 97, 99, 123 topological space, 48, 64, 96 topologically isomorphic, 248, 250, 251, 259 topology, 39, 64, 228, 232 topology generated by a collection of sets, 66 total variation, 52 totally bounded, 58 totally disconnected, 62 totally ordered set, 305 transcendental number, 24, 192 translation, 117, 203 transpose, 121 triangle inequality, 10, 13, 20, 34, 102 trigonometric polynomial, 85, 245 trigonometric series, 238 Tychonoff’s theorem, 70 uniform boundedness principle, 81, 109 uniformly absolutely continuous, 215 uniformly bounded sequence of functions, 213 uniformly continuous function, 51, 57 uniformly convergent sequence, 46, 50, 58, 82, 134, 244 unimodular group, 231, 232, 235, 265 union, 273 unipotent matrix, 204, 225 unit ball, 37, 111 unit circle, 22, 238 unit group, 95 unit interval, 304 unit sphere, 165 unit vector, 337 unitarily equivalent, 235, 267 unitary character, 239, 246, 247, 265 unitary dual, 234, 267 unitary equivalence, 234, 270 unitary group, 125 unitary operator, 131, 233, 238 unitary representation, 232, 233, 267 units in a ring, 82, 87 universal set, 273 unramified extension, 95 upper bound, 2, 306 upper triangular matrices, 344 upper triangular unipotent matrices, 344 Urysohn’s lemma, 68, 189 Urysohn’s Metrization Theorem, 68 usual metric on Rn, 34 valuation, 89, 90 variation, 52 vector space, 256, 313 Vitali Convergence Theorem, 215 volume of a rectangle, 194, 197 wave equation, 238 weakly continuous representation, 233 weakly convergent sequence, 134 Weierstrass’s theorem, 82 Weil, Andr´ e, 77 well-defined, 287 well-ordered set, 305 well-ordering principle for Z, 283 Zahlen, 280 Zygmund, Antoni, 237

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