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Index Abel–Weierstrass lemma, 413 basic absolute condition number, columns, 65 302 variables, 63 addition basis, 13 closure of, 5, 576 standard, 13, 51 adjoint, 121 Bauer–Fike theorem, 560 adjugate, 73, 76 Beppo Levi theorem, 337 algebraic multiplicity, 143 Bernstein almost everywhere, 329, 332 operator, 82 convergence, 333 polynomials, 54 equal, 332 Bessel’s inequality, 97 nonnegative, 340 bijection, 38 analytic function, 411, 414, 434 bijective, 38, 637 angle preserving, 131 bilinear, 89, 667 arclength, 383 block matrix function, 384 inverse, 666 parametrized by, 384 multiplication, 665 argument principle, 445 Bochner integral, 320 arithmetic-geometric mean inequality, Boolean ring, 575, 595 118 boundary of a set, 192 Arnoldi bounded basis, 531 above, 645 method, 547 functions, 113 asymptotic behavior, 561 linear transformation, 114, 186, automorphism 200, 235 of a ring, 595 linear transformation theorem, 213, of a vector space, 38 216 axiom of choice, 647 set, 197 Brouwer fixed-point theorem, 225, 277 Bézier curve, 54 back substitution, 61 calculus Banach space, 211 fundamental theorem of, 403 Banach-valued regulated integral cancellation, 588 multivariable, 324 canonical epimorphism, 41, 600 single-variable, 230 Cantor barycentric diagonal argument, 650 Lagrange interpolation, ternary set, 332 610 cardinality, 648 weight, 610 Cartesian product, 14, 19, 596, 630 679 680 Index Cauchy computer aided design, 58 differentiation formula, 427 condition, 302 integral formula, 413, 424 condition number sequence, 195, 196, 263 of a matrix, 305 Cauchy–Goursat theorem, 419, 421 absolute, of a function, 302 Cauchy–Riemann eigenvalue problem, 307 equation, 410 of a matrix, 307 equations, polar form, 451 relative, of a function, 303 theorem, 410 congruent Cauchy–Riemann equation, 410 modulo n, 631 Cauchy–Schwarz inequality, 91 modulo an ideal, 598 Cayley–Hamilton theorem, 479 connected, 179, 224 chain, 20, 647 component, 396 chain rule conservative vector field, 388 for holomorphic functions, 412 continuity Fréchet derivative, 259 uniform, 199 change of basis, 53 continuous change of variables, 263, 351 at a point, 185 characteristic polynomial, 143 function, 185, 188 Chinese remainder linear extension theorem, 213, 216 problem, 602 Lipschitz, 186, 200, 235, 254, 278 theorem, 80, 601, 604 pointwise, 187 classical adjoint, 76 uniformly, 199 closed continuously differentiable, 266 n-interval, 321 contour, 416 ball, 191 integral, 416, 417 set, 190 simple closed, 419 closure contraction of a set, 192 mapping, 278 of operations, 5, 576 mapping principle, 279 codomain, 635 mapping principle, uniform, 282 cofactor, 73 mapping, uniform, 282 expansion, 65, 75 convergence column space, 126 absolute, 212 commutative almost everywhere, 333 ring, 575 linear, 286 commutative diagram, 638 of sums, 212 compact, 179, 203 pointwise, 210 support, 122 quadratic, 286 complementary uniform, 210, 413 subspace, 16 uniform on compact subsets, 263 complete, 198 convex, 261 completion, 361 coordinate complex change of, 299 conjugate, 654 coordinates numbers, 653 change of, 48, 351 composition, 36, 636 hyperbolic, 360 of continuous functions, 187 hyperspherical, 356 Index 681 in a basis, 47 differentiable polar, 352 complex function, 408 spherical, 354 continuously, 253 coset, 23, 598 function, 241, 246, 252 operations, 24, 599 dimension, 18 countable, 648 formula, 46 cover direct sum, 14, 16 open, 203 Dirichlet function, 332 Cramer’s rule, 77, 217 divides, 585 cross product, 667 division property, 583 CRT, see Chinese remainder theorem domain curl, 402 Euclidean, 583 curve of a function, 635 differentiable, 242 dominated convergence theorem, 342 fitting, 129 dot product, 89 piecewise-smooth, 382 Drazin inverse, 500, 501 positively oriented, 398 dual space, 248 simple closed, 381 smooth, 382 Eckart–Young, Schmidt, Mirsky smooth parametrized, 381 theorem, 167 smooth, oriented, 382 EEA, see extended Euclidean algorithm, smooth, unoriented, 382 587 cutoff phenomenon, 564 eigenbasis, 151 eigennilpotent, 483 Daniell integral, 320, 328 eigenprojection, 463, 475 data compression, 168 eigenspace, 140 De Morgan’s Laws, 630 generalized, 465, 468, 486 decay matrix, 564 eigenvalue, 140 decomposition semisimple, 492 LU, 62 simple, 307, 496 polar, 165 eigenvector QR, 103 generalized, 468 singular value, 162 elementary Wedderburn, 500 matrix, 59 dense, 190 product, 68 derivative, 242 empty set, 627 directional, 244 equivalence higher, 266 modulo a subspace, 22 linearity, 256 class, 22, 598, 632 of a complex function, 408 modulo n, 21, 631, 633 of a parametrized curve, 242 modulo an ideal, 598 second, 266 relation, 598, 631 determinant, 65 Euclidean De Moivre’s formula, 655 algorithm, 586 diagonal matrix, 162 domain, 573, 583 diagonalizable, 151 extended algorithm, 587 orthonormally, 157 Euler’s formula, 411, 415, 655 diffeomorphism, 349 extension by zero, 325 682 Index extension theorem, 18, 29 Gauss’ mean value theorem, 426 exterior of a simple closed curve, 397 Gauss–Seidel extreme value theorem, 205 convergence, 524 method, 522 Fatou’s lemma, 340 gcd, 586 field, 659 generalized finite intersection property, 206 eigenspace, 465, 468, 486 first isomorphism theorem Heine–Borel theorem, 208 for rings, 600 inverse, see pseudoinverse for vector spaces, 42 Leibniz integral rule, 348 fixed point, 278 minimal residual method (GMRES), fomula 530 Euler, 411 step function, 379 formal power series, 576 geometric multiplicity, 141 Fourier transform, discrete, 622 Givens rotation, 545 Fréchet derivative, 246, 252 GMRES, 530, 533 chain rule, 259 Google, 498 higher-order, 266 Gram–Schmidt, 99 product rule, 257 modified, 101 real finite-dimensional case, 246 graph, 635 Fredholm greatest alternative, 136 common divisor, 586 integral transform, 281 element, 646 free variables, 63 lower bound, 645 Frobenius Green’s theorem, 381, 399 inner product, 90, 136 Fubini’s theorem, 320, 344 Hölder’s inequality, 118 for step functions, 372 Heine–Borel theorem function, 635 generalized, 208 analytic, 411, 413 on Rn, 204 entire, 408 Hermitian conjugate, 84, 664 holomorphic, 408 Hessian matrix, 266 integrable (Daniell–Lebesgue), 330 Hilbert’s identity, 471 regulated integrable, 324, 375 Hilbert–Schmidt Riemann integrable, 320, 325, 340 inner product, 90 smooth, 266 holomorphic fundamental function, 408, 434 subspaces theorem, 124 open mapping theorem, 445, 450 theorem of algebra, 430, 591 homeomorphism, 222 theorem of arithmetic, 590 Householder, 105 theorem of calculus, 262 Householder transformation, 105, 106 theorem of calculus for line hyperbolic coordinates, 360 integrals, 388 hyperplane, 321 fundamental theorem hyperspherical coordinates, 356 of calculus, 403 ideal, 578 Gamma function, 359 generating set, 581 Garner’s formula, 606 identity map, 39 Index 683 ill conditioned, 303 irreducible image, 594, 635 element, of a ring, 588 implicit function theorem, 294 matrix, 497 index of a matrix, 466 isolated point, 189 index set, 639 isometric embedding, 362 indicator function, 228, 323 isometry, 116 induced isomorphic rings, 596 metric, 208 isomorphism norm on linear transformations, of rings, 595, 596 114 of vector spaces, 36, 38 norm, from an inner product, 111 iterated integral, 345 induction, 644 iterative infimum, 645 numerical methods, 519 inherited metric, 208 solvers, 521 injective, 637 iterative methods, 520 inner product, 88 Frobenius, 90, 136 Jacobi method, 522 Hilbert–Schmidt, 90 convergence, 524 positivity of, 91 Jacobian space, 89 determinant, 297 standard, 89 matrix, 249 integers, 628 Jordan integrable canonical form, 140, 506 function, Daniell–Lebesgue, 330 curve theorem, 396 function, regulated, 324 Jordan normal form, 480 function, Riemann, 320, 325, 340 Kantorovich, 315 function,regulated, 375 kernel, 35, 594 on an unbounded domain, 337 Kreiss constant, 562 integral Kreiss matrix theorem, 563 Bochner, 320 Kronecker delta, 95 contour, 417 Krylov Daniell, 320, 328 basis, 527 iterated, 345 methods, 526 Lebesgue, 320, 328 solution, 529 mean value theorem, 262 subspace, 506, 527 interior of a set, 182 Lagrange of a simple closed curve, 397 basis, 603 interior point, 182 decomposition, 602 intermediate value theorem, 225 interpolant, 603 intersection of sets, 630 interpolant, barycentric form, invariant subspace, 147, 462, 468 610 inverse function theorem, 298 interpolation, 610 invertible, 38 interpolation, barycentric, 610 element, 578 Lagrange–Hermite interpolation, 616, left, 642 617 right, 642 Lanczos method, 548 involution, 172 Laplace equation, 145 684 Index Laurent system, 32, 58, 105, 519, 520, 527, expansion, 436 530 polynomial, 576 homogeneous, 35, 40 series, 433, 436 overdetermined, 127 series, principal part, 438 transformation, 31–33 law of cosines, 92 bounded, 114 leading entry, 61 composition of, 51 least matrix representation, 49 element, 647 Liouville’s theorem, 430 upper bound, 645 Lipschitz upper bound property, 646 continuous, 186, 200, 235, 254, least squares, 127 278 Lebesgue locally at a point, 254 integrable, 367 lower bound, 645 integral, 320, 328 l.u.b. property, see least upper bound number, 206 property left LU decomposition, 62 eigenvector, 154 invertible, 642 manifold, 381 pointing normal vector, 398 map, 635 Legendre polynomial, 101, 453 Markov chain, 564 Leibniz integral rule matrix, 663 generalized, 348 -norm, 116 1 Leibniz’s integral rule, 346 1-norm, 116 Leontief, Wassily, 520 augmented, 60 level set, 191, 293, 635 block, 665 limit decay, 564 inferior, 340 diagonalizable, 151 of a function, 188 diagonally dominant, 524 of a sequence, 193 Hermitian conjugate, 664 point,
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