362 INDEX

definition, 332volume, 332

partial derivatives, 147, 155continuous, 155

partial order, 60partially ordered set, 60partition of unity, 84partitioned matrix, 11permutation, 35permutation matrix, 18permutation symbol, 334pi systems, 184piecewise smooth

differentiable, 265, 274pivot column, 27point of density, 241pointwise compact, 78pointwise convergence, 77pole, 298Polish space, 68, 235polynomial

in many variables, 117polynomials, 117positive linear functional

measure, 193positive part, 207power set, 49preserving distance, 16primitive, 297product measure, 223product of matrices

inverse, 24products of Borel sets, 196

Radon measure, 191Radon Nikodym

theorem, 220Radon Nikodym derivative, 220rank, 29rank of a matrix, 44rank theorem, 173rational function, 117real and imaginary parts, 207regular, 191

measure space, 232regular values, 302removable singularity, 299retract, 226

fixed point property, 226right handed system, 329right inverse, 31–33right polar factorization, 15rot, 336row equivalent, 27row operations, 18row operations

linear relationships, 25row rank, 44row reduced echelon form, 26row reduced echelon form

existence, 27uniqueness, 28

Russell’s paradox, 51

saddle point, 134Sard’s theorem, 244, 304scalar field, 336scalars, 91Schroder Bernstein theorem, 52Schur’s theorem, 13second derivative, 151second derivative test, 171, 172sections of open sets, 154separable metric space

Lindeloff property, 68separated sets, 80sequence, 64

Cauchy, 65subsequence, 65

sequentially compact set, 69series

double sum, 56set

F sigma, 191sets, 49

G delta, 191sgn, 33

uniqueness, 35sigma algebra, 179sign of a permutation, 35signed measure, 218

Hahn decomposition, 218signed measures

Hahn decomposition, 218simple functions

approximation, 180