rudin

INDEX 341 Interior point, 32 Mapping, 24 Open mapping, 100, 223 Intermediate value, 93, 100, 108 affine, 266 Open set, 32 Intersection, 27 continuous, 85 Order, 3, 17 Interval, 31, 302 continuously differentiable, 219 Into, 24 linear, 206 lexicographic, 22 Inverse function, 90 open, 100, 223 Ordered field, 7, 20 Inverse function theorem, 221 primitive, 248 Inverse image, 24 uniformly continuous, 90 k-tuple, 16 Inverse of linear operator, 207 (See also Function) pair, 12 Inverse mapping, 90 set, 3, I8, 22 Invertible transformation, 207 Matrix, 210 Oriented simplex, 266 Irrational number, 1, 10, 65 product, 211 Origin, 16 Isolated point, 32 Orthogonal set of functions, 187 Isometry, 82, 170 Maximum, 90 Orthonormal set, 187, 327, 331 Isomorphism, 21 Mean square approximation, 187 Outer measure, 304 Mean value theorem, 108, 235 Jacobian, 234 Measurable function, 310 Parameter domain, 254 Measurable set, 305, 310 Parameter interval, 136 Kellogg, 0. D., 281 Measurable space, 310 Parseval's theorem, 191, 198, 328, Kestelman, H., 167 Measure, 308 Knopp, K.. 21, 63 331 outer, 304 Landau, E.G. H., 21 Measure space, 310 Partial derivative, 215 Measure zero, set of, 309, 317 Laplacian, 297 Partial sum, 59, 186 Least upper bound, 4 Mertens, F., 74 Partition, 120 property, 4, 18 Metric space, 30 of unity, 251 Lebesgue, H.\\L., 186 Minimum, 90 Lebesgue-integrable function, 315 Mobius band. 298 Perfect set, 32 Lebesgue integral, 314 Monotone convergence theorem, Periodic function, 183, 190 Lebesgue measure, 308 318 7T, 183 Lebesgue's theorem, 155, 167, Monotonic function, 95, 302 Plane, 17 318,321 Monotonic sequence. 55 Poincare's lemma, 27 5, 280 Left-hand limit, 94 Multiplication (see Product) Pointwise bounded sequence, 155 Leibnitz, G. W., 71 Negative number, 7 Pointwise convergence, 144 Negative orientation, 267 Polynomial, 88 Length, 136 Neighborhood, 32 L'Hospital's rule, 109, 113 Newton's method, 118 trigonometric, 185 Limit, 47, 83, 144 Nijenhuis, A., 223 Positive orientation, 267 Niven, I.. 65, 198 Power series, 69,172 left-hand, 94 Nonnegative number, 60 Primes, 197 lower, 56 Norm, 16, 140, 150, 326 Primitive mapping, 248 pointwise, 144 right-hand, 94 of operator, 208 Product, 5 subsequential, 51 Nor111al derivative, 297 upper, 56 Normal space, 101 Cauchy, 73 Limit function, 144 Normal vector, 284 of complex numbers, 12 Limit point, 32 Nowhere differentiable function, of determinants, 233 Line, 17 Line integral, 255 154 of field elements, 5 Linear combination, 204 Linear function, 206 Null space, 228 of forms, 258, 260 Linear mapping, 206 Nutt vector, 16 of functions, 85 Linear operator, 207 Number: inner, 16 Linear transformation, 206 of matrices, 211 Local maximum, 107 algebraic, 43 of real numbers, 19, 20 Localization theorem, 190 cardinal, 25 scalar, 16 Locally one-to-one mapping, 223 complex, 12 Logarithm, 22, 180 decimal, 11 of series, 73 Logarithmic function, 180 finite, 12 irrational, 1, 10, 65 of transformations, 207 Lower bound, 3 negative, 7 Projection, 228 nonnegative, 60 Proper subset, 3 Lower integral, 121, 122 positive, 7, 8 Lower limit, 56 rational, I Radius, 31, 32 real, 8 of convergence, 69, 79 McShanc. E. J ., 3 I3 One-to-one correspondence. 25 Range, 24, 207 Onto, 24 Rank, 228 Open cover, 36 Rank theorem, 229 Ratio test, 66 Rational function, 88 Rational number, I Real field, 8 Real line, 17 Real number, 8 Reitl part. 14

342 INDEX Rearrangement, 75 Set, Sum, of linear transformations, 207 Rectifiable curve, 136 dense, 9, 32 of oriented simplexes, 268 Refinement, 123 elementary, 303 of real numbers, 18 Reflexive property, 25 empty, 3 Regular set function, 303 of series, 59 Relatively open set, 35 finite, 25 Remainder, 211, 244 independent, 205 of vectors, 16 Restriction, 99 infinite, 25 Summation by parts, 70 Riemann, B., 76, 186 Support, 246 Riemann integral, 121 measurable, 305, 310 Supremum, 4 Riemann-Stieltjes integral, 122 nonempty, 3 Supremum norm, 150 Riesz-Fischer theorem, 330 open, 32 Surface, 254 Right-hand limit, 94 ordered, 3 Symmetric difference, 305 Ring, 30 I perfect, 32, 41 Robison, G. B., 184 relatively open, 35 Tangent plane, 284 Root, IO uncountable, 25, 30, 41 Tangent vector, 286 Root test, 65 Set function, 30 I Tangential component, 286 Row matrix, 217 u-ring, 30 I Taylor polynomial, 244 Taylor's theorem, I I0, 116, 176, 24. Saddle point, 240 Simple discontinuity, 94 Scalar product, 16 Simple function, 313 Thorpe, J. A., 280 Schoenberg. I. J., 168 Simplex, 247 Schwarz inequality, 15, 139, 326 Thurston, H. A., 21 Segment, 31 affine, 266 Torus, 239-240, 285 Self-adjoint algebra, 165 differentiable, 269 Total derivative, 213 Separable space, 45 oriented, 266 Transforqiation (see Function; Separated sets, 42 Singer, I. M., 280 Separation of points, 162 Solid angle, 294 Mapping) Sequence, 26 Space: Transitivity, 25 compact metric, 36 Triangle inequality, 14, 16, 30, 140 bounded, 48 complete metric, 54 Trigonometric functions, 182 Cauchy, 52, 82,329 connected, 42 Trigonometric polynomial, 185 convergent, 47 of continuous functions, 150 Trigonometric series, 186 divergent, 47 euclidean, 16 double, 144 Hilbert, 332 Uncountable set, 25, 30, 41 of functions, 143 of integrable functions, 315, 326 Uniform boundedness, 155 measurable. 310 Uniform closure, 151 increasing, 55 measure, 310 Uniform continuity, 90 monotonic, 55 metric, 30 Uniform convergence, 147 normal, IOI Uniformly closed algebra, 161 pointwise bounded, 155 separable, 45 Uniformly continuous mapping, 90 pointwise convergent, 144 Span, 204 Union, 27 Sphere, 272, 277, 294 Uniqueness theorem, 119, 258 uniformly bounded, 15 5 Spivak, M., 272, 280 Unit cube, 247 Square root, 2, 81, 118 Unit vector, 2 I7 uniformly convergent, 157 Standard basis, 205 Upper bound, 3 Standard presentation, 257 Upper integral, 121, I 22 Series, 59 Standard simplex, 266 Upper limit, 56 Stark, E. L .. 199 absolutely convergent, 71 Step function, 129 Value, 24 alternating, 71 Stieltjes integral, 122 Variable of integration, 122 Stirling's formula, 194, 200 Vector, 16 convergent, 59 Stokes' theorem, 253, 272, 287 Vector field, 281 divergent, 59 Stone-Weierstrass theorem, 162, Vector space, 16, 204 Vector-valued function, 85 geometric, 61 190,246 nonabsolutely convergent, 72 Stromberg, K., 21 derivative of, 112 power, 69, I72 Subadditivity, 304 Volume, 255, 282 product of, 73 Subcover, 36 trigonometric, 186 Subfield, 8, 13 Weierstrass test, 148 uniformly convergent, 157 Subsequence, 51 Weierstrass theorem, 40, 159 Set, 3 Subsequential limit, 51 Winding number, 20 I at most countable, 25 Subset, 3 Borel, 309 Zero set, 98, 117 bounded, 32 dense, 9, 32 Zeta function. 14 l bounded above, 3 proper, 3 Cantor, 41, 81, 138, 168, 309 closed, 32 Sum, 5 compact, 36 complete orthonormal, 331 of complex numbers, 12 connected, 42 of field elements, 5 convex, 31 of forms, 256 countable, 25 ot· functions, 85