464 INDEX

integraldecreasing function, 163functions in L1, 169linear, 170

integral domain, 380integral over a measurable set, 175interchange order of integration, 135interior point, 40intermediate value theorem, 73intersection, 3intersection of two curves, 212intervals

notation, 3invariance of domain, 210invariant subspace, 438inverse Fourier transform, 274inverse function theorem, 106, 108inverse image, 5, 52inversions, 416irreducible, 21

relatively prime, 22isolated singularity, 393

Jacobiperiodic functions, 426

James map, 319Jensens inequality, 239Jordan arc, 74Jordan curve

finite length, 453Green’s theorem, 457orientation, 457

Jordan curve theorem, 213

Lagrange multipliers, 111Laplace transform, 268, 272

inverse well defined, 274inversion, 279, 404properties, 272

Laurent series, 385, 387principal part, 387

least squares regression, 110Lebesgue

integral desires to be linear, 168integral of nonnegative function,

164integral simple function, 165measurable a.e. equal to Borel

measurable, 197

measure, 156number, 47other definitions of integral, 167points, 235

Lebesgue decomposition, 184level curves

intersecting at right angles, 376lim inf, 30

properties, 32lim sup, 30

properties, 32limit of a function, 54limit of a sequence

well defined, 39limit point, 40limits

existence of limits, 30Lindeloff property, 46line integrals

fundamental theorem, 128Linear

not continuous, 303linear maps

closed, 308continuous, 70equivalent conditions, 70

linear operators, 66linear transformation

norm, 66linear transformations, 66Liouville’s theorem, 373Lipschitz mappings of sets of measure

zero, 216little o notation, 94locally integrable, 285locally one to one, 379logarithm

analytic, 350branch, 350branches, 350principal branch, 350

Lusin, 220Lyapunov Schmidt procedure, 113

matrix of linear transformation, 66maximal chain, 9maximal function

Radon measures, 235