452 INDEX

inner product space, 304inner regular, 189integral

decreasing function, 159functions in L1, 165linear, 166

integral domain, 371integral over a measurable set, 171interchange order of integration, 132interior point, 38intermediate value theorem, 71intersection, 3intersection of two curves, 208intervals

notation, 3invariance of domain, 206invariant subspace, 427inverse Fourier transform, 268inverse function theorem, 104, 106inverse image, 5, 50inversions, 406irreducible, 22

relatively prime, 22isolated singularity, 384

Jacobiperiodic functions, 416

James map, 313Jensens inequality, 235Jordan arc, 72Jordan curve

finite length, 441Green’s theorem, 445orientation, 445

Jordan curve theorem, 209

Lagrange multipliers, 109Laplace transform, 263, 266

inverse well defined, 268inversion, 273, 394properties, 266

Laurentseries, 377

Laurent series, 377, 378principal part, 378

least squares regression, 108Lebesgue

integral desires to be linear, 164

integral of nonnegative function,160

integral simple function, 161measurable a.e. equal to Borel

measurable, 193measure, 151number, 45other definitions of integral, 163points, 231

Lebesgue decomposition, 180level curves

intersecting at right angles, 370lim inf, 30

properties, 31lim sup, 30

properties, 31limit of a function, 52limit of a sequence

well defined, 37limit point, 38limits

existence of limits, 30Lindeloff property, 44line integrals

fundamental theorem, 126linear

not continuous, 297linear maps

closed, 302continuous, 68equivalent conditions, 68

linear operators, 64linear transformation

norm, 64linear transformations, 64Liouville’s theorem, 366Lipschitz mappings of sets of measure

zero, 212little o notation, 92locally integrable, 278locally one to one, 371logarithm

analytic, 344branch, 344branches, 344principal branch, 344

Lusin, 216Lyapunov Schmidt procedure, 111