358 INDEX OF TERMINOLOGY normed linear space, 64, 102, 103, 116, 119, 219 nowhere dense set, 80 one-to-one, 292 one-to-one correspondence, 292 onto, 292 open ball, 20, 36 open cover, 17, 56 open interval, 73 open map, 66, 96, 99, 106 open mapping, 112 Open Mapping Theorem, 110, 111 open rectangle, 181, 188, 194 open set, 16, 21, 38, 48, 65, 198, 200 order axioms, 281 order isomorphic, 3 order of a group, 342 order of an element, 342 order topology, 73 ordered field, 2, 3, 42, 73, 86, 289 ordered integral domain, 2, 72, 73, 289 ordered pair, 277 ordinary least squares approximation, 168 orthogonal, 335, 337 orthogonal complement, 129, 140 orthogonal group, 123, 124, 344 orthogonal matrix, 204 orthogonal projection, 235 orthogonal set, 132 orthogonal transformation, 347 orthonormal basis, 132, 133, 140, 245, 258, 337 orthonormal set, 132, 133, 245 Osgood, William Fogg, 1 Ostrowski’s theorem, 102 Other Hahn-Banach Theorem, 113–115, 117 outer measure, 196, 197, 228, 229 outer regular Borel measure, 200 pairwise disjoint, 274 paraboloid, 165 parallelepiped, 333 parallelogram law, 126, 128 Parseval’s identity, 245, 249 partial derivative, 150, 154, 161, 162 partial differential equations, 238 partial sum, 26 partially ordered set, 305 partition, 185–188 perfect set, 80 permutation, 325 perpendicular, 337 Picard’s theorem, 79 piecewise continuous, 244 piecewise continuous function, 241, 243, 244 piecewise smooth function, 241, 243, 244, 264, 265 pigeonhole principle, 292 Plancherel’s theorem, 248, 249, 256, 257 plane, 333 point, 332 pointwise convergent sequence, 46, 51, 244 pointwise evaluation, 109 polar form, 23 polynomial function, 82, 296 Pontryagin-van Kampen Duality Theorem, 248, 250 positive definite bilinear form, 334 positive definite Hermitian form, 126–128 positive definite quadratic form, 168, 169 positive operator, 135, 136 power set, 279 preimage, 294 prime number, 285 primitive root of unity, 23, 92 principal branch of the argument, 23 principal units, 91 product measure, 224, 232 product rule, 152 product topology, 69, 70 projection, 69, 336 projection onto a closed subspace, 129, 131 projection onto the k-th coordinate, 151 proper subset, 272 pure imaginary, 19 quadratic form, 168, 169 quantifiers, 273 quotient field, 73 quotient map, 98, 99 quotient topology, 70, 266 radius of convergence, 29, 158, 163 ramified extension, 95 range, 292 rapidly decreasing at infinity, 248, 255 ratio test, 28 rational number, 101, 286

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