INDEX OF TERMINOLOGY 357 least upper bound property, 12, 25, 42, 72, 74, 76 Lebesgue inner measure, 195, 226 Lebesgue integrable, 213 Lebesgue integral, 208, 212, 224 Lebesgue measurable set, 199, 202 Lebesgue measure, 192, 194, 200, 204, 229, 249 Lebesgue measure zero, 199–202, 205 Lebesgue outer measure, 194, 195, 226 Lebesgue’s theorem, 182, 184 left Haar measure, 227–232 left regular representation, 235 left-hand derivative, 241 limit, 10, 13, 20, 44 limit infimum, 54, 55 limit supremum, 54, 55 line, 333 linear combination, 314 linear functional, 112 linear independence, 315 linear isometry, 117 linear isomorphism, 320 linear operator, 319 linear transformation, 106, 149, 150, 202, 319 linearly dependent, 316 linearly independent set, 132, 133, 179 Liouville’s theorem, 148 Lipschitz condition, 79, 202 Little Hahn-Banach Theorem, 112 local maximum, 145, 148, 166–170, 172, 173 local minimum, 145, 148, 166–170, 172, 173 locally compact, 89, 123 locally compact abelian group, 232, 246–248, 265 locally compact group, 123, 124, 126, 194, 265 locally compact space, 70 locally compact topological group, 227 locally compact topological space, 99 locally constant function, 248, 261 lower bound, 2, 306 lub, 306 Lusin’s theorem, 208 mathematical induction, 283 matrix, 121, 123, 154, 155, 162, 169, 179, 230, 268, 322 maximal compact subgroup, 126 maximal element, 306 maximal ideal, 87, 89, 95 maximum value, 144, 166, 174 Mazur-Ulam theorem, 118, 119, 203 Mean Value Theorem, 145, 148, 149, 156–161, 163, 177, 180 measurable, 196 measurable function, 192, 205, 206, 208, 209, 212 measurable set, 196 measure, 192, 193, 260 measure space, 193, 205, 208 measure zero, 182, 185, 187–189, 207 metric, 34, 86, 228 metric induced by the norm, 103 metric space, 34, 65, 199, 205 metric topology, 228 Michelle function, 293 minimum value, 144, 166, 174 Minkowski’s inequality, 219 Minkowski’s integral inequality, 221–223 mixed partial derivative, 160, 161 model, 310 monomorphism, 343 Monotone Convergence Theorem, 210, 211, 221, 222 monotonic function, 176, 182 monotonic increasing function, 52, 182 monotonic increasing sequence, 14 monotonic sequence of functions, 58 monotonicity of Lebesgue outer measure, 195 monotonicity of measure, 194 monotonicity of the integral, 209 motion group, 266 mutlinomial coeﬃcient, 162 nearly continuous function, 182 negative definite quadratic form, 168 neighborhood, 9, 65, 96, 97 Nested Intervals Theorem, 9 non-Archimedean absolute value, 94 non-Archimedean triangle inequality, 85, 87, 92 nonmeasurable set, 204 nonnegative extended real numbers, 193 nonoverlapping intervals, 51 nonsingular linear transformation, 176–179, 202 norm, 102, 127, 334 normal subgroup, 98, 126, 231, 344 normal topological space, 67, 68 normalized vector, 337

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