INDEX 361

lower triangular, 43right inverse, 43right, left inverse, 43row, column, determinant rank, 44upper triangular, 43

matrixblock multiplication, 11inverse, 23, 31more columns than rows, 29non zero kernel, 29partitioned, 11Schur’s theorem, 13

matrix multiplicationblock, 11

maximal chain, 60maximal function

Radon measures, 239maximum modulus theorem, 300mean value theorem

Cauchy, 124measurability

limit of simple functions, 181measurable, 186

complex valued, 207equivalent formulations, 180linear combinations, 207

measurable complex functionssimple functions, 210

measurable functions, 179pointwise limit, 179simple functions, 180

measurable into (−∞,∞], 180measurable sets, 186measure, 182

inner regular, 191outer regular, 191properties, 183

measure on a manifold, 266measures

decreasing sequences of sets, 183increasing sequences of sets, 183regularity, 235

measures from outer measures, 187metric, 63

properties, 63metric space, 63

compact sets, 70complete, 66

completely separable, 67open set, 63separable, 67

metric tensor, 340, 342metric tensor, 351min max theorem, 134min-max theorem, 133minor, 41mixed partial derivatives, 156Mobeus band

graph, 292mollifier, 250monotone convergence theorem, 204multi - index, 117multi-index, 152multi-index notation, 152Muntz theorem, 139

negative part, 207Newton’s second law, 346no retract onto boundary of ball, 313non equal mixed partials

example, 158norm

p norm, 98normal, 257, 275Normed linear space, 98

open ball, 63open set, 63

open cover, 68open mapping theorem, 300open set, 63open sets

countable basis, 67relative, 261

orderedpartial, 60totally ordered, 60

orientable, 292oriented atlas, 264orthonormal, 100outer measure, 185

measurable, 186outer regular, 191

G delta and F sigma sets, 192

p norms, 99parallelepiped