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Author Index

AAbel, N. H., 81, 333, 760Alaoglu, L., 522Alekseev, V. B., 760Alexándrov, P. S., 307, 311, 790Apostol, T.M., 342Archimedes, 339Argyros, S. A., 615Aristotle, 73Arzelà, C., 53, 332Ascoli, G., 53, 332Asplund, E., 590Auerbach, H., 517

BBaire, R., 53, 54, 216Banach, S., 53, 132, 329, 507, 521, 526, 612Barrow, I., 356Beals, R., 594Bendixson, I. O., 308Benyamini, Y, 333, 585, 593, 612, 615, 707Bernoulli, D., 64Bernoulli, J., 99Bernstein, F., 31Bessaga, Ch., 522Bishop, E. A., 532Blass, A., 630Boas, R. P., Jr., 779Bochner, S., 775Bolzano, B., 62, 71, 140, 230, 329Bonic, R., 591Borel, É., 50, 122, 262Borsuk, K., 53Borwein, J., 591Brouwer, L. E. J., 53, 517, 522Brown, J., 333Bruckner, A. M., 294

Bruckner, J. B., 294

CCantor, G., 21, 35, 37, 53, 126, 157, 361Carathéodory, C., 115, 519, 820Cardano, G., 333Carleson, L., 475Cauchy, A. L., 62, 72, 140, 168, 250, 363, 453Cech, E., 584Cellérier, Ch., 230Cesàro, E., 476Chebyshev, P., 497, 764Christensen, J. P. R., 831Cohen, P., 38Conrad, B., 756Cousin, P., 745

Dda Vinci, L., 104Dalí, S., 14Darboux, J. G., 155, 211, 213, 214, 342Davies, R. O., 376de la Vallée Poussin, Ch. J., 9De Morgan, A., 2Dedekind, R., 19, 21, 621del Ferro, S., 333Denjoy, A., 428, 790Descartes, R., 3Deville, R., 219, 646, 712Diestel, J., 774, 805, 811Dieudonné, J., 57Dini, U., 196, 222, 470Diofanto, 9Dirichlet, J. P. G. Lejeune, 9, 59, 138du Bois-Reymond, P., 230, 475Dugundji, J., 517, 638

© Springer International Publishing Switzerland 2015 835V. Montesinos et al., An Introduction to Modern Analysis,DOI 10.1007/978-3-319-12481-0

836 Author Index

EEdison, T. A., 631Edwards Jr., C. H., 540, 807, 808Egorov, D., 227Ekeland, I., 585Enflo, P., 564, 577Euclid, 8, 9Euler, L., 53, 100, 455, 659

FFabian, M., 300, 310, 311, 521, 522, 524, 532,

539, 544, 558, 582, 584, 591, 593, 612,615, 643, 646, 730, 811, 814, 823, 826,827

Fatou, P., 399Fejér, L., 476Fermat, P. de, 9Ferrari, L., 333Fischer, E. S., 562Folland, G. B., 779Fomin, S. V., 301Fontana, N., see TartagliaFourier, J., 455Fréchet, M., 53, 533, 597Frampton, J., 591Fubini, G., 91, 479, 785

GGödel, K., 38Gâteaux, R., 533Galileo Galilei, 32, 109Gauss, C. F., v, 9, 333, 503Gibbs, J., 4Godefroy, G., 219, 646, 712Goffman, C., 19Goursat, E., 756Gowers, W. T., 521, 613–615Gregory, J., 356Grothendieck, A., 524Guirao, A. J., 812

HHölder, O., 453Hájek, P., 300, 310, 311, 521, 522, 524, 532,

539, 544, 558, 582, 584, 591, 593, 612,615, 643, 646, 730, 811, 814, 823, 826,827

Haar, A., 567Habala, P., 300, 310, 311, 521, 522, 524, 532,

539, 544, 558, 582, 584, 591, 593, 612,615, 643, 646, 730, 811, 814, 823, 826,827

Hadamard, J. S., 9, 250Hahn, H., 527

Hake, H., 790Hardy, G. H., 60, 225Hausdorff, F., 53, 370Haydon, R. G., 615Heine, H. E., 50, 157Helly, E., 700Henstock, R., 790Hermite, Ch., 758Hewitt, E., 280, 626Hilbert, D., 34, 35, 53, 550

JJames, R. C., 532Jameson, G. J. O., 236, 584, 734Jensen, J., 449Johnson, W. B., 612Jordan, C., 204, 472

KKönig, J., 31Kadets, M. I., 522, 544, 730Kamke, E., 637Kant, I., 1Katznelson, Y., 800Kelley, J. L., 581, 582, 630Kepler, J., 177Kirk, W. A., 333Knaster, B., 214, 681Knuth, D. E., 652Kolmogorov, A. N., 301, 475Komorowski, R., 615Krein, M.G., 519Kronecker, L., 5, 517Kuratowski, K., 53, 214, 681Kurzweil, J., 591, 790

LL’Hôspital, G. de, 173López-Pouso, R., 423Lagrange, J. L., 168, 333Landau, E. G. H., 697Laplace, P. S., 100, 612Le Corbusier, 14Leach, B., 592Lebesgue, H., 109, 115, 196, 199, 413, 428Legendre, A. M., 9Leibniz, G., 1, 30, 375Levi, B., 397Lin, P. K., 593Lindenstrauss, J., 333, 544, 574, 585, 590, 591,

593, 612, 615, 707Lipschitz, R., 209Loève, M., 503

Author Index 837

Looman, H., 790Lu, J., 415Luzin, N. N., 193, 194, 475

MMarkov, A., 497Mascheroni, L., 659Maurey, B., 614, 615Mazur, S., 53, 588Mazurkiewicz, S., 329Mertens, F., 98Miller, A. W., 739Milman, D. P., 519Milman, V. D., 613, 615Minkowski, H., 452, 519, 820Montesinos, V., 300, 310, 311, 521, 522, 524,

532, 539, 544, 558, 582, 584, 591, 593,612, 615, 643, 646, 730, 811, 814, 823,826, 827

Morse, H. C. M., 207Myint-U, T., 483

NNatanson, I.P., 217Newman, D. J., 9Newton, I., 249, 279, 356

OOdell, T., 615Oxtoby, J. C., 330

PPacioli, L., 14Palais, R. S., 586Pełczynski, A., 522Peano, G., 311, 363, 617Pelant, J., 300, 310, 311, 521, 522, 524, 532,

539, 544, 558, 582, 584, 591, 593, 612,615, 643, 646, 730, 811, 814, 823, 826,827

Perron, O., 428, 790Pettis, B. J., 774Phelps, R., 532Picard, É., 367Piranian, G., 795Plato, 13Poincaré, H., 53Poisson, S. D., 500Preiss, D., 590, 591Pringsheim, A., 92Pythagoras, 10, 12

RRademacher, H. A., 591

Ramsey, F. P., 645Raphson, J., 249Riemann, B., 9, 85, 175, 340, 342, 348, 413Riesz, F., 238, 516, 558, 562Rolle, M., 167Ruffini, P., 760Russell, B., 57

SSard, A., 207Schauder, J., 564, 593Schechter, E., 790Schlumprecht, Th., 615Schröder, E., 31Schur, A., 811Schwartz, L., 594Schwarz, K. H. A., 453Sierpinski, W., 637Sims, B., 333Smale, S., 586Smith, H. J. S., 361Šmulyan, J. L., 537Snobar, M. G., 544Sobczyk, A., 544Steinhaus, H., 53, 125, 330, 544Sternberg, S., 167Sternfeld, Y., 593Stieltjes, T. J., 342Stirling, J., 654Stolz, O., 92Stone, M. H., 584Straszewicz, S., 819Stromberg, K. R., 41, 193, 280, 447, 625, 626,

654, 672, 675

TTakagi, T., 230Tarski, A., 132Tartaglia, 333Taylor, B., 240Thim, J., 230Thomson, B. S., 294Tietze, H. F. F., 294Tolias, A., 615Tomczak-Jaegermann, N., 615Torunczyk, H., 730Tsirelson, B. S., 613, 615Tychonoff, A. N., 581Tzafriri, L., 544, 574, 615

UUrysohn, P. S., 311

838 Author Index

VValdivia, M., 756van der Waerden, B. L., 230Vanderwerff, J., 591Vitali, G., 115, 130, 196Volterra, V., 53, 361von Neumann, J., 38

WWallis, J., 108, 783Weierstrass, K., 71, 140, 153, 224, 230, 235Weyl, H., 310Whitehead, A. N., 1Whitfield, J. H. M., 592

Widom, H., 395, 479Wiles, A., 9Wright, E. M., 60

ZZahorski, Z., 795Zajícek, L., 590Zeno, 73Zermelo, E. F. F., 629Zizler, V., 219, 300, 310, 311, 521, 522, 524,

532, 539, 544, 558, 582, 584, 591, 593,612, 615, 643, 646, 712, 730, 811, 814,823, 826, 827

Zorn, M., 132, 629

General Index

0-hyperplane, 510

A(a.e.), 121absolute continuity

of a function, see function, absolutelycontinuous

of the measure defined by an integral, 407absolute value

of a real number, 25of an integer, 6

accumulation point, 288, 653algebraic basis, see basis, Hamelalmost everywhere, see (a.e.)angle between two vectors, 553antiderivative, 356, 356, 357, 359, 360, 376,

378–380, 428, 486, 662, 692, 746, 750,751, 768

absolutely continuous, 409, 422and integration by parts, 379and the Fundamental Theorem of Calculus,

358, 360calculus of —, 756differentiable, 357function without —, 752, 756Lipschitz, 355of the Takagi–van der Waerden function, 683uniqueness, 360

approximate identity, 687, 688, 689and orthonormal basis in L2[0, 2π ], 565and the Dirichlet problem, 829and Weierstrass approximation theorem,

233, 689argument of a function, 3arithmetic progression, 69

sum of the terms of —, 658

associated real normed space, 547, 547, 549,550

linear functionals, 548to C, 547

asymptote, 143, 690, 690Auerbach basis, see basis, Auerbachaverage

limit of —, 656of a function, 340, 341, 345, 347of the partial sums of a Fourier series, 476,

568velocity, 168

Axiom of Choice, 130, 132, 629and cardinality, 639and Vitali’s Lemma, 131, 132

BB(2)

has isometric copy of ∞, 810not separable, 810

Baire classes, 195, 217, 217, 218, 219, 417,645, 711

Baire space, 308, 328, 328characterization, 328if complete metric, 328if Polish, 329

Baire spaces, 54ball

closed, 286open, 286

Banach space, 297, 443, 507, 507, 516, 519,523, 584, 612–615, 731, 801, 802, 804,806, 809, 815, 816, 820, 826, 827

Asplund, 590, 591complex, 547dual, 511finite-dimensional, 512, 519, 520, 535, 814

© Springer International Publishing Switzerland 2015 839V. Montesinos et al., An Introduction to Modern Analysis,DOI 10.1007/978-3-319-12481-0

840 General Index

has an Auerbach basis, 518hereditarily indecomposable, 615infinite-dimensional, 521, 613, 811, 823, 827linearly isomorphic to Hilbert, 544locally uniformly rotund, 827not linearly isomorphic to Hilbert, 574, 823product, 559reflexive, 532, 532separable, 522, 582, 730, 812, 814strictly convex, 532subreflexive, 532without Schauder basis, 564

Banach–Tarski paradox, 132base

of a positional system, 14, 15, 18, 21of a positive fraction xq , 12of a topology, 193, 194, 303

countable, 303base expansion, 21–23, 37, 74, 79, 634, 741

binary, 137, 635, 642, 664, 681finite, 16, 16, 22

characterization, 58periodic, 16, 16, 22terminating, 16, see base expansion, finiteternary, 129, 188

base representation, see base, of a positionalsystem

basisalgebraic, see basis, HamelAuerbach, 517, 517, 518, 519, 543

and projections, 544dual, 809Haar, 567Hamel, 513, 514, 516, 518, 524, 809, 810

existence, 630extension, 526

orthonormal, see orthonormal basisSchauder, 564, 564, 812

in p , 812in c0, 812

Bauer maximum principle, 796Bernoulli’s inequality, see inequality, BernoulliBernstein set, 646, 669, 669binary system, 4binomial coefficients, 102, 102, 255, 262, 263,

265, 276, 279, 279, 280, 281, 499, 500,633, 633, 634, 654, 677, 715, 721

binomial distribution, 499, 499, 500, 501, 503,504

binomial expansionfinite, 102, 142, 279, 633, 654, 677, 721

binomial integral, 764binomial series, 279, 280, 280, 281

biorthogonal system, 517, 518, 524Borel set, 122, 122, 124, 189

inverse image by a continuous function, 674is measurable, 122, 195

boundary, 43, 43, 46, 167, 442, 520, 531, 808and continuity of the characteristic function,

675characterization, 44, 725Dirichlet problem, 828power series, 257set with empty —, 43, 651

boundary conditionsof a partial differential equation, 829

B(X,Y ), 510, 525, 526, 545, 549, 575

Cc0, 507, 525, 607, 773, 774, 802, 803, 806, 823

ball not compact, 525canonical basis, 522closed unit ball not w-compact, 573closed unit ball not pointwise compact, 827compact operator from —, 814compactness in —, 812dual space of —, 523, 816extreme points, 817fixed points, 593not complemented in ∞, 544not hereditarily indecomposable, 614not isomorphic to 2, 522, 823not isomorphic to C[0, 1], 816not isomorphic to Hilbert, 544not reflexive, 525not Schur, 826operator from —, 814operator into —, 814Rainwater theorem, 826Schauder basis, 812separable, 306space without copies of —, 613, 615uniformly continuous function of the unit

sphere of —, 615c00(�), 294, 508, 806

and the Banach–Steinhaus theorem, 545and the open mapping theorem, 521discontinuous linear functional, 806not complete, 300separable, 306

C[0, 1], 286, 315, 330, 332, 507, 510, 512, 581,725, 729, 730, 734, 803

and Auerbach basis, 519closed unit ball, 288, 299

not compact, 316Polish, 312separable, 305

General Index 841

compact operator, 813compact set in —, 741dual space, 512, 728–730

nonseparable, 512equivalent norm, 822extreme points, 817Fréchet differentiable function, 820is Banach, 298, 507, 568is Polish, 311norm 2-rough, 592norm no uniform limit of Fréchet

differentiable functions, 592not Fréchet bump, 591not isomorphic to c0, 816not isomorphic to Hilbert, 544, 823separable, 305, 733

c0(�), 294, 294, 300, 306, 508, 812is Banach, 300separability, 306

Cantor diagonal process, 35, 38, 133, 332, 523,582, 638, 701, 743

Cantor scheme, 307, 308Cantor space, 127Cantor ternary set, 127, 209, 741

and Lebesgue singular function, 186, 211,404

into a perfect Polish space, 308is nowhere dense, 128is perfect, 128of positive measure, 129, 361, 415, 417, 668,

711, 754, 788cardinal number, 637

finite, 638ordering, 638power, 640product, 640product of two —, 640representative of a —, 637sum, 640transfinite, 638

cardinalitylarger than, 30, 638larger than or equal to, 30, 638smaller than, 30, 638smaller than or equal to, 30, 638two sets with the same —, 30, 637

Cartesian product, 3, 3, 136, 314, 624, 640Cauchy product, 98, 98, 265, 276Cauchy–Schwarz inequality, see inequality,

Cauchy–Schwarzchain, 303, 319

in a preordered set, 628characteristic function, 405

characteristic function of a set, 138, 138, 304,306, 352, 393, 403, 404, 411, 414, 417,419, 567, 600, 640, 642, 644, 667, 700,711, 713, 746, 753, 754, 774, 812

Baire class, 217, 711continuity, 675, 711lower semicontinuity, 709measurability, 189, 405, 786Riemann integrability, 352, 788

Chernoff’s bound, 498closed ball centered at x, 507closed unit ball, 507closure, 43, 43, 45, 46, 287, 289

characterization, 44, 45, 68, 287, 725, 727contains the infimum, 651contains the supremum, 651is closed, 43of a bounded set, 44of a convex set, 550, 801of a product of sets, 630of a totally bounded set, 319of open ball, 725

cluster point of a sequence, 71, 71, 287, 657,727

characterization, 71codomain

of a function, 3combinatorial numbers, see binomial

coefficientscompact set, 47, 48, 48, 154, 325

Arzelà–Ascoli theorem, 332ball of a finite-dimensional space, 515ball of Hilbert in w-topology, 573characterization, 320, 322, 326C∞-function on —, 687closed and bounded interval is —, 48closed subset of —, 49continuous function on —, 153, 157, 324continuous image of a — 153, 323, 324continuous one-to-one function on —, 154,

679convex, 582, 593, 818

extreme points, 819, 820convex hull, 520, 523dual unit ball in w-topology, 584equicontinuous family of functions on —,

325in c0, 812in R

characterization, 50, 72has maximum and minimum, 50

in closed convex hull of a sequence, 524in dual of a Banach space, 522

842 General Index

in finite-dimensional spacecharacterization, 519convex, 519

infinitehas accumulation point, 50

is bounded, 49, 313is closed, 49, 313is complete, 313Lebesgue number, 325metric, 312, 740

continuous image of Cantor space, 311,583

nearest point, 740then separable, 313

nested sequence, 72product of —, 582, 630product of —, 313scattered, 737

characterization, 737then totally bounded, 318

compactificationStone–Cech, 584

complement of a set, 1completeness

axiom, 624not a topological notion, 724of R, see R, complete

completion, 301, 313, 327construction, 301, 735of a metric space, 301, 738of Riemann integrable functions, 388unique, 301

composition of two functions, 137connected set, 51, 155, 256, 261, 306, 520,

528, 652, 675, 691, 745continuous image of a —, 152countable metric, 738in R

characterization, 52R is —, 52, 651, 652

continued fraction, 311, 643, 643Continuum Hypothesis, 37, 38, 638contraction, 335, 336–338, 367, 368, 370, 742

fixed point of a —, 336fixed point of a —, 335, 336without fixed point, 742

convergencealmost everywhere, 215in measure, 237, 237, 238, 792

characterization, 791in the sense of distributions, 602mean square, 482pointwise, 215, 220

uniform, 220, 222Abel test, 225Dirichlet test, 225implies pointwise, 220

convexcombination, 439, 441, 744, 792hull, 439, 521, 792

characterization, 792closed, 801of compact, 519, 523

set, 439, 530, 637, 805characterization, 439, 792closure, 529compact, 582, 593, 818, 820distance to —, 556exposed points, 818, 823extreme points, 817–819separation, 528

convex setclosure of a —, 801

convolution, 604continuity, 479Dirichlet kernel, 466, 479, 481Fejér kernel, 477of periodic distributions, 607, 827of two continuous functions, 233, 607, 687,

689of two integrable functions, 479

corollaryPalais–Smale, 586

cover, 47Vitali, 197

CP [0, 2π ], 568, 595, 605, 829critical point, 166, 167, 693

Morse–Sard theorem, 167not extreme, 166

cutDedekind, see Dedekind cutgolden, see golden cut

DDarboux condition (D), 773, 773Darboux property, see Intermediate Value

PropertyDe Morgan formulas, 2, 631decimal system, 4decreasing

function, see function, decreasingsequence of sets, 118

Dedekind cut, 19, 20, 621negative, 622nonnegative, 622ordering of —, 20positive, 20, 622

General Index 843

rational, 20, 621definite integral, 359δ-net, 303, 319

countable, 303, 304finite, 317, 318, 321, 523

�-system, 646root of the —, 646

density of a set at a point, 786derivative, 156, 160, 167, 168

and extrema, 165as the slope, 161bounded, 209, 210Dini, 196, 198, 199, 200directional, 533, 534, 535discontinuous, 211Fréchet, 533, 590Gâteaux, 533, 588has intermediate value property, 167higher order, 164, 164, 166, 239, 240, 242,

243, 246, 258, 259and extrema, 171

Lebesgue integrable, 401left-hand-side, 199not continuous, 213not Riemann integrable, 361of a constant function, 163of a distribution, 601of a monotone function, 199of a polynomial, 163, 173of a series, 252of inverse function, 184one-sided, 160partial, 534positive, 170right-hand-side, 199sequence of —, 261zero (a.e.), 428

devil’s staircase, see function, Lebesguesingular

diameter of a set, 55, 286differential

equation, 273, 338, 363, 367, 611, 691, 828of a function, 682

properties, 171of a function at a point, 162unique, 162

dimension of a vector space, 513Dini derivative, see derivative, Dinidirect sum, 559Dirichlet function, 138, 149, 150, 163, 415,

651, 673, 675, 747Henstock–Kurzweil integral, 746is Baire class 2, not Baire class 1, 711

is discontinuous everywhere, 149Lebesgue integrable, 396, 416limit of a sequence of Riemann integrable

functions, 403limit of Riemann integrable functions, 373measurable, 193no limit at any point, 140not Baire class 1, 195, 218not Riemann integrable, 353

Dirichlet kernel, 464, 465, 465, 466, 477, 607,770

and convolution, 479, 481and partial sums of Fourier series, 466integral of —, 466, 468norm of —, 800properties, 466

disjoint sets, 2distance, 53, 283

between two points, 23, 25, 725between two spheres, 612, 614defined by a norm, 297, 506Euclidean, 283from a closed to a compact set, 125from a point to a hyperplane, 532, 806from a point to a subset, 289, 307, 556, 556,

561, 676, 691, 708, 711, 727, 803Lipschitz, 289not attained, 732

supremum, 285, 710distribution

of a continuous random variablenormal, 503, 801

of a discrete random variablebinomial, see binomial distribution, 499Poisson, 500two point, 498, 499

of natural numbers, 11of prime numbers, 9periodic, see periodic distribution

distribution function, 501, 502of a continuous random variable, 501, 501,

502, 504of a discrete random variable, 490, 800

division algorithm, 6, 15divisor

of an integer number, 7domain of a function, 3, 135dot product, see inner product

seeinner product, 283dual space, see space, dual

Eeigenvalue, 574, 807, 809

and point spectrum, 577

844 General Index

compact operator without —, 577finite-dimensional space, 574, 807left-shift operator, 825right-shift operator, 577

eigenvector, 574, 807emptyset, 1equicontinuous, 324equicontinuous at a point, 324equivalence class, 620Euler formulas, 567Euler number e, 664

as exp 1, 264as a limit, 101as a product, 108as a sum, 103, 104digits, 104is irrational, 103

Euler–Mascheroni constant, 659event, 488

elementary, 488eventually, 63expected value of a random variable, see

random variable, meanexponential

function, see function, exponentialgrowth, 273system, 461, 564, 796

exposed point, 818, 823in B1 , 819in Hilbert, 823

extended real number system, 24extreme point, 817, 818, 820, 824

convergence on —, 826Gδ , 818if exposed, 818of B1 , 817of BC[0,1], 817of Bc0 , 817of BH , 817of BR2 , 808of w-compact convex, 819of closed convex hull, 520of closed convex in finite-dimensional space,

521of compact convex, 524, 807, 819of compact convex in finite-dimensional

space, 519, 796, 818of face, 817

extremumglobal, 153, 155, 165, 165, 167, 168, 449,

693, 728, 729, 807, 808, 819, 824local, 165, 165, 166, 168, 246, 449, 692, 693

criteria for —-, 171

necessary condition, 165strict, 246

Ffamily

cover, 47of all finite subsets, 3of all subsets, 2of nonoverlapping intervals, 206of sets, 2open cover, 47pairwise disjoint, 2

Fejér kernel, 476, 476and averages of partial sums of Fourier

series, 477properties, 476

Fermat conjecture, 9Finite Induction Principle, 4, 617, 632, 635,

649Finite Intersection Property, 326, 326first category, see set, first categoryfirst element

in a preordered set, 628fixed point, 333, 333, 334, 368–370

compact convex in finite-dimensional space,593

continuous function on [0, 1], 334function without —, 593, 742function without —, 334if f [k] a contraction, 336of f continuous on compact convex, 593of a contraction, 335, 336, 370of a limit, 338of a strictly metric function, 337

Fourier coefficients, 464, 471, 475, 482, 483,564, 604, 611, 798

exponential form, 461null sequence, 464, 607of a periodic distribution, 602, 603–605of an even function, 462of an odd function, 462of derivative, 568trigonometric form, 461with respect to an orthonormal basis, 563

Fourier expansion, see Fourier seriesFourier integral, 482Fourier series, 455, 563, 797–799

(a.e.) convergence, 475Cesàro convergence, 477divergent, 474, 475exponential form, 461, 462of a δ-function, 607of a periodic distribution, 603, 603of a test function, 605

General Index 845

of an odd function, 799partial sum, 464, 465, 475, 476, 482, 607pointwise convergence, 467, 469–472, 798,

799Dini condition, 471Jordan condition, 473

trigonometric form, 461uniform convergence, 478, 568, 603, 605,

799Fourier transform, 486fraction, 10

proper, 10fractional part, 26, 636, 653frequently, 63Fresnel integral, 771Fσ -set, see set, FσFunction

Lebesgue measurablecharacterization, 774

functionabsolute value, 25absolutely continuous, 206, 206, 207–209,

211, 213, 221, 355, 409, 422, 423, 428,429, 446, 480, 565, 568, 666, 703, 705,706, 786, 793, 794

and integration by parts, 429and null sets, 207and the Fundamental Theorem of Calculus,

422, 423, 789composition, 706composition with Lipschitz, 704if convex, 446if Lipschitz, 209image of a measurable set, 207non-Lipschitz, 211not Lipschitz, 210product of —, 703then of bounded variation, 206then uniformly continuous, 206with zero derivative then constant, 207

absolutely continuous part, 428affine, 441Baire class 0, 710Baire class 1, 217, 219, 227, 417, 645, 710,

711, 712continuous at Gδ , 217not Baire class 0, 217

Baire class 2, 217, 218, 711not Baire class 1, 217, 218

bijection, see function, bijectivebijective, 3Bochner integrable, 775Borel measurable, 189

bounded, 139bounded variation, 201, 202–205, 209, 211,

213, 342, 421, 422, 428, 444, 446, 473,474, 485, 502, 702, 705, 782, 786

not absolutely continuous, 209characterization, 205if absolutely continuous, 206if Lipschitz on compact interval, 209if monotone, 202then (a.e.) differentiable, 205, 401

bump, 262, 591, 674, 684–687, 821, 822,825

ceiling, 26codomain, 3concave, 440continuous, 147, 148, 149, 151, 152,

167–170, 189, 213, 290, 672–676, 694,700, 704, 708, 726, 740

algebraic properties, 151approximated by polynomials, 235at a point, 147, 290Baire class 0, 216characterization, 150, 151, 290, 597extension, 294, 676, 679fixed point, 334Fourier series of a —, 477, 799has antiderivative, 357if convex on open interval, 442if differentiable, 162if uniformly —, 157if uniformly —, 290Intermediate Value Property, 155inverse of —, 154, 183, 679is a distribution, 598modulus of continuity, 158nonmeasurable image, 196not absolutely continuous, 209not differentiable, 163not of bounded variation, 202not uniformly —, 158, 322, 678nowhere differentiable, 162, 230, 329, 330nowhere monotone, 231, 690on a compact convex, 593on a compact space, 153, 157, 323, 324one-sided, 147, 444, 502, 674one-to-one, 517property (D), 773then Borel, 191then Darboux, 155then Riemann integrable, 353, 773then upper function, 392with divergent Fourier series, 474, 475without fixed point, 334

846 General Index

convex, 439, 440, 506, 792–794, 796, 802boundedness, 442characterization, 449, 793continuity, 442, 588differentiability, 443–446, 535, 588–590,

795differentiable, 444extrema, 448Fréchet differentiable, 589if affine, 441locally Lipschitz, 442, 445, 445, 587monotonicity, 444one-sided-derivatives, 442, 443stability, 440test, 448test for —, 447three-chord property, 441

convex continuousthen absolutely continuous, 446

Darboux, 211, 214decreasing, 139defined by Lebesgue integral

is absolutely continuous, 409defined by Riemann integral

is absolutely continuous, 355density, 503devil’s staircase, see function, Lebesgue

singulardifferentiable, 160

Fréchet, 533Gâteaux, 533then continuous, 162

Dirichlet, see Dirichlet functiondomain, 3, 135even, 139exponential, 253, 263, 264, 264, 265–268,

272, 273, 450, 461, 595, 626, 721analytic, 264complex, 627Euler formulas, 567, 626infinitely differentiable, 264properties, 264

extension, 135, 294floor, 26gauge, 745hyperbolic cosine, 274hyperbolic sine, 273hyperbolic tangent, 274increasing, 139infinitely differentiable, 258, 259, 261, 262,

267, 275, 278, 432, 595, 686–688, 829divergent Taylor series, 259if power series, 251, 255, 260

if real analytic, 260not real analytic, 260not Taylor series, 258

injective, 3integrable on a measurable set, 406inverse, 137Lebesgue integrable, 394, 396, 400, 403,

450, 749, 776, 777, 797characteristic function, 405characterization, 781derivative of bounded variation function,

401Fourier coefficients, 463Fourier integral, 482Fourier series, 458, 461, 462, 467, 475,

479if Riemann integrable, 417indefinite integral, 429integral zero, 406is a distribution, 598Lebesgue points, 791not Riemann integrable, 419product, 395, 404properties, 395then Henstock–Kurzweil integrable, 745

Lebesgue singular, see Lebesgue singularfunction

Lipschitz, 209, 210, 211, 213, 221, 289, 290,355, 356, 363–365, 428, 445, 446, 558,603, 704, 708–710, 733, 741, 798

and Fundamental Theorem of Calculus,782

at a point, 706extension, 707Fourier series, 478, 568, 799Fréchet differentiability, 591on large distances, 707oscillation stable, 613product, 708sufficient condition, 210then absolutely continuous, 209without fixed point, 593

locally Lipschitz, 210, 442, 445, 470, 535,587

and Fourier series, 470logarithm, 267, 273, 386, 721

and prime numbers, 9computational device, 272infinitely differentiable, 267properties, 267, 722Taylor polynomial, 241Taylor series, 269, 270

lower semicontinuous, 709

General Index 847

is Baire class 0 or 1, 710on a metric space, 708characterization, 709

measurable, 189, 237, 293, 388, 389, 392,395, 402, 403

absolute vale, 192approximation by continuous function, 194Borel, 189, 191characteristic function, 405characterization, 191–194composition, 191Lebesgue, 189properties, 189sequence of —, 227, 237, 238, 399, 401step, 390

midpoint convex, 447, 447, 794negative part, 26Newton integrable, 746nowhere differentiable, 176odd, 139one-to-one, see function, injectiveoscillation stable, 613periodic, 178pointwise Lipschitz, 210, 704, 706, 707polynomial, see polynomialpositive part, 26range, 135rational, 139, 757, 761, 764, 765

proper, 757, 759real analytic

at a point, 260, 260, 261on an interval, 260, 260, 261, 264

restriction, 135Riemann, see Riemann functionRiemann integrable, 358

characterization, 345, 346, 349composition of two —, 415equally spaced partitions, 347properties, 350without antiderivative, 358Σ-measurable step, 192singular, 428, 428singular part, 428smooth, see function, differentiablestep, 358, 390strictly concave, 440, 695strictly convex, 432, 433, 439, 440, 450, 452,

695, 796characterization, 449test, 448

strictly decreasing, 139strictly increasing, 139strictly metric, 336

Takagi–van der Waerden, 230, 230, 231–233,683, 755

uniformly continuous, 157, 157, 158–160,181, 211, 213, 290, 323, 324, 615,677–680, 704, 788

algebraic properties, 159characterization, 158, 676, 679extension, 679, 728if absolutely continuous, 206if bounded monotone, 680if Lipschitz, 290not bounded variation, 212not Lipschitz, 290, 706space of —, 733then continuous, 157then Lipschitz on large distances, 707

vector-valued Riemann integrable, 772functional, see linear functional

Ggauge, 745gauge integral, see integral, Henstock–KurzweilGδ-set, see set, Gδgeneral interval, 390geometric progression, 70golden cut, 14, 14, 636golden ratio, 636

not rational, 636graph of a function, 3, 136greatest common divisor, 7

HHölder’s inequality, see inequality, HölderHaar wavelets, 567Hamel basis, see basis, Hamelheat equation, 610Hermite method for primitives, 758Hilbert cube, 567Hilbert space, 456, 467, 478, 482, 505, 511,

526, 532, 543, 544, 550, 551, 554, 554,555, 556, 558–561, 565, 567, 569, 597,815, 823

all functionals attain their norm, 570bidual space, 823bounded operators in —, 816characterization, 615compact operators in —, 815complemented subspaces, 574differentiability, 574, 825differentiability of the norm, 569distortable, 615dual space, 572, 573, 816, 823duality mapping, 571, 573exposed points, 823

848 General Index

extreme points, 817, 823Fourier series, 563has an orthonormal basis, 562no equivalent rough norm, 825null sequences in –, 826orthonormal basis in —, 560, 812reflexive, 572separable, 560

all linearly isometric, 562all linearly isomorphic, 563Fourier series, 563orthonormal basis in —, 562, 564, 569

series in —, 823strictly convex, 533, 553, 571, 817supporting hyperplane, 570weak compactness, 573

Hilbert’s hotel, 34homeomorphism, 53, 195, 291, 292, 517, 576,

584, 710, 739and compactness, 724and completeness, 300and total boundedness, 317, 322characterization, 291equivalent metrics, 292, 293, 300, 305, 307,

317, 322, 700, 724R, open interval, 292

hyperplane, 510, 510, 522, 543, 613distance from a point, 532, 806linear isometry onto an —, 521separating, 520, 528, 530, 550space not isomorphic to —, 521, 615support, 519, 520, 530, 530, 531, 533, 536,

537, 539, 543, 549, 569, 570, 796tangent, 531, 549

Iimaginary part

of a complex functional, 548immediate predecessor, 5immediate successor, 5increasing

function, see function, increasingsequence of functions, 222sequence of sets, 118

indefinite integral, 355, 409indicator function, see characteristic function

of a setinequality

Bernoulli, 64, 101Bessel, 562, 568Cauchy–Schwarz, 395, 453, 454, 551, 552,

554, 568, 571, 606, 696, 796Chebyshev, 498

Hölder, 453, 804Jensen

finite form, 449, 453integral form, 450

Markov, 497Minkowski, 452, 731

integral, 555triangle, 26, 78, 284, 293, 335, 506, 506,

507, 514, 552, 557, 707, 709, 723, 725,738, 802, 804

infimum, 622, 651infimum of a bounded above set in R, 27infinite product, 91, 105, 105

convergent, 105characterization, 105necessary condition, 106sufficient conditions, 106

divergent, 105product of the —, 105terms of an —, 105Wallis formula, 108, 783

inflection point, 695, 695initial segment, 647inner product, 459, 550, 551, 551, 554, 572,

573Euclidean, 551, 560in dual space, 572integral, 554norm associated to an —, 551series, 554

inner product space, 283, 454, 550, 551, 551,552, 553, 823

complete, see Hilbert spacethen strictly convex, 553

input for a function, 3integer

even, 7negative, 6nonnegative, 6odd, 7positive, 6

integer part, 26integral

Dirichlet, 435Henstock–Kurzweil, 745, 745, 746Lebesgue, see Lebesgue integralNewton, 746of probability, 784Riemann, see Riemann integral

interior, 43Intermediate Value Property, 155, 155, 156,

213, 214, 277, 334, 358, 371, 372, 675,680, 681, 802

General Index 849

and root of polynomials, 681continuous function has —, 156discontinuous function having the —, 156,

681of the derivative, 167, 445, 746

intersection of sets, 2interval

bounded, 24closed, 24general, 52is uncountable, 56left half open, 24length of an–, 25nested —, see theorem, Nested Intervalsnondegenerate, 25open, 24right half open, 24

invariant subspace, 577compact operators have —, 577

isometry, 291, 291, 301, 736, 740antilinear, 569

Hilbert to dual, 571–573linear, 511, 521, 810, 816

adjoint, 801characterization, 511Hilbert to 2, 562into bidual, 531

isomorphismantilinear, 572Hilbert —, 573linear, 511, 513, 559, 574, 575, 583, 613,

731if closed range, 546if one-to-one and onto, 545, 576, 822, 826in finite-dimensional space, 516, 545, 577,

811

Jjump discontinuity, 149, 150, 420

function with only —, 748function without —, 156of a monotone function, 186, 205, 213of the derivative, 364, 443, 445, 690

Kkernel

Dirichlet, see Dirichlet kernelFejér, see Fejér kernelof a linear functional, 528, 529, 806of an integral equation, 369of an integral transform, 485, 486Poisson, 829

Kronecker delta, 513, 517

LL’Hôspital’s Rule, 173, 186, 245, 386, 387,

433, 466, 468, 662, 682, 683, 690, 716,721, 748, 753

L1

complete, 7771, 605, 774L1[a, b], 3882, 810L[a, b], 479Lagrange multiplier method, 540, 807, 808Landau notation, 697Laplace integral, see probability integralLebesgue constants, 800Lebesgue integral, 206, 362, 387, 394, 457,

775, 779, 790, 862and improper Riemann integral, 420, 772extension of Riemann integral, 417Fundamental Theorem for the —, 423, 781implies Henstock–Kurzweil, 745of a characteristic function, 405of a Lebesgue integrable function, 394of a step function, 390of an upper function, 392parametric, 430properties, 395

Lebesgue measurable set, 114, 115, 189, 501,665, 667

Borel set is —, 122cardinality of the family of —, 644characteristic function, 405characterization, 122, 123difference of two —, 125, 668not Borel, 195open (closed) set is —, 121regularity, 118, 122, 125sequence of —, 133set not —, 130, 669σ -algebra, 118

Lebesgue measure, 121, 389, 405, 501, 665,667

inner, 124countable union, 112

outer, 110is translation invariant, 114of an interval, 111regularity, 113

Lebesgue number of a covering, 325Lebesgue outer measure, 665Lebesgue points, 790Lebesgue property, 774Lebesgue set, 790

850 General Index

Lebesgue singular function, 186, 187, 195,196, 209, 211, 213, 221, 403

and Fundamental Theorem of Calculus, 422,428

bounded variation, 209continuous, 187differentiable (a.e.), 187increasing, 187is singular, 428not a N -function, 211not absolutely continuous, 209Riemann integral, 374, 403second class not first class, 209

Leibniz criterion on alternating series, 84, 84,253, 269, 270, 384, 435, 661, 720, 748

lemmaAbel’s partial summation, 81, 82, 83, 226,

457Bezout, 7, 8Borel–Cantelli on sequences of measurable

sets, 133, 238Cousin on tagged partitions, 745Dirichlet on fractions, 59, 59, 61, 653Riemann–Lebesgue, 463, 464, 468, 470,

473, 484, 607Riesz on finite-dimensional spaces, 516,

517, 549Root, 646sandwich, 181Sierpinski on subsets of N, 643Vitali on nonmeasurable sets, 130Zorn, 303, 319, 629

lengthof an interval, 25

limes inferior, see limit inferiorlimes superior, see limit superiorlimit

(a.e.), 215double, 91, 91in the sense of distributions, 602, 610inferior, see limit inferioriterated, 91, 91of a function, 140, 146, 653, 671

at ∞, 143Cauchy criterion, 147one-sided, 143, 143, 144uniqueness, 140

of a sequence, 62, 286, 657infinite, 64properties, 65uniqueness, 63

of a sequence of functions

pointwise, see pointwise limit, of asequence of functions

uniform, see uniform limit, of a sequenceof functions

of a sequence of ordinal numbers, 649, 650of a sequence of Riemann integrable

functions not Riemann integrable, 373,403

of a sequence of sets, 132of test functions, 595, 607superior, see limit superior

limit inferior, 656characterization, 656of a function, 144, 671of a function on a metric space, 708of a sequence, 67, 656, 657of a sequence of sets, 132, 669

limit ordinal, see ordinal number, limitlimit point, see accumulation pointlimit superior

characterization, 656of a function, 144, 145, 149, 218, 590, 592,

671, 681of a function on a metric space, 709of a sequence, 67, 68, 70, 80, 81, 85, 87,

251, 255, 259, 655–657of a sequence of functions, 190of a sequence of sets, 132, 133, 669

line segment between two points, 439linear combination, 513linear functional, 509, 513

and hyperplanes, 510, 806attaining its supremum, 532continuous, 509, 820discontinuous, 512, 516, 805, 806extending a continuous —, 526, 542extension of a continuous —, 527imaginary part, 548not attaining its norm, 525real part, 548space of all —, 511, 513, 515space of all continuous —, 511supporting a set, 530, 569, 570, 817

linear space, see vector spaceL∞(E), 294, 294

is Banach, 300is not separable, 306

Lipschitz constant, 209locally uniform convergence of a sequence, 717logarithm

decimal, 273natural, 273

lower bound, 27, 622

General Index 851

p , 293, 732complete, 299distortable, 615dual of —, 539not q , 732not isomorphic to Hilbert, 544, 823Schauder basis in —, 812separable, 305space without copies of —, 613, 615

Lp(E), 293, 306, 454, 567is Banach, 300

np , 731

Mmajorant function, 401map, see mappingmapping

antilinear, 571decreasing, 628increasing, 628, 646linear, 509open, 545strictly decreasing, 628strictly increasing, 628

maximal elementof a preordered set, 628

maximumlocal, 165of a function, 153, 155

on a compact set, 153of a subset of R, 50

maximumpointwise — of two functions, 235

meanarithmetic, 450, 451geometric, 450, 451of a random variable, see random variable,

meansquare convergence, 482value for integration, see theorem, First

Mean Value for Riemann Integrationvalue theorem, see theorem, Lagrange Mean

Valuemeasure

absolutely continuous, 408, 408characterization, 408

and integration, 405convergence in —, see convergence, in

measure, 791countably additive, 121defined by integral, 408, 778inner, see inner measureLebesgue, see Lebesgue measure

of angles, 18, 177outer, see outer measureprobability, 488, 488, 501, 501, 503

product, 496median of a random variable, see random

variable, medianMersenne primes, 633metric, 53

associated to a norm, 506bounded, 286compatible with a topology, 303discrete, 285equivalent, 292, 302, 307, 723, 724, 731

not completeness, 724Hausdorff, 370, 742

complete, 742metric space, 53, 151, 283, 284, 286, 288, 289,

324bounded, 286closure in a —, 289δ-separated, 303homeomorphic, 291isometric, 291isometric to subset of ∞(�), 301Lindelöf, 303perfect, 308separable, 302

characterization, 303subspace is —, 304

subspace is —, 285minimum

local, 165of a function, 153, 155

on a compact set, 153of a subset of R, 50

Minkowski’s inequality, see inequality,Minkowski

mode of a random variable, see randomvariable, mode

modulus of continuity, 158, 158, 159, 677, 679

Nneighborhood, 42Newton integral, see integral, NewtonNewton’s method for finding roots, 249, 271Newton–Raphson method, see Newton’s

method for finding rootsnorm, 284, 506

associated to an inner product, 551Day’s, 805dual, 511, 569, 572equivalent locally uniformly rotund, 827locally uniformly rotund, 827no equivalent C1, 821

852 General Index

no equivalent Fréchet, 822no equivalent rough on Hilbert, 825operator —, 525rough, 592, 592ε—, 592

strictly convex, 532, 819supremum

on ∞(�), 285on BC(M), 584on C[0, 1], 288, 512, 582on c0, 507on c00, 508on X∗, see norm, dual

normed linear space, see normed spacenormed space, 506

complete, see Banach spacenorms

equivalent, 508, 515, 803, 804in every finite-dimensional space, 514sufficient condition, 822

two non-equivalent norms, 804nowhere dense, 54nuber

primeMersenne, 633

numberalgebraic, 643

cardinality, 643composite, 8irrational, 12, 23, 29, 58, 60, 61, 73, 103,

125, 137, 149, 163, 176, 195, 219, 297,445, 635, 642, 643, 651, 653, 668, 670,673, 675, 739, 773, 774, 795

approximation by —, 654cardinality, 36dense, 36, 37, 45expansion, 22, 23

prime, 8there are infinitely many —, 9

rational, 10, 19, 20, 73, 176, 620–622, 634,654

approximation by —, 59, 68, 653cardinality, 34dense, 36, 37, 45expansion, 16, 22ordering, 11power of —, 12

real, 20numbers

congruent modulo p, 35dyadic, 23relatively prime, 7

Oopen ball centered at x, 507open cover, 47operator, 509

bounded, see operator, continuouscompact, 524compact non finite-rank, 525continuous

characterization, 509, 510finite-rank, 524in finite-dimensional space, 511invertible, 575left shift, 825left shift on 2, 521Lipschitz, 509norm, 510on finite-dimensional space, 515right shift, 577right shift on 2, 521

orderlinear, see order, totalof a distribution, 598total, 628well —, 628

N has a —, 5, 618order isomorphism, 6order topology, 624ordered field, 624ordinal number, 646

fundamental sequence, 647generalized sequence, 647initial segment, 647limit, 648, 649limit —, 647ordering, 647power, 649product, 649product of two —, 648sequence of first —, 649sum of —, 648

orthogonal, 459, 553, 567basis, 812, 815complement, 553projection, 815to a subset, 553trigonometric system, 459, 460vectors, 553, 578

orthonormal, 459orthonormal basis, 459, 560, 563, 567, 815,

823countable, 560, 564dual, 573every Hilbert space has —, 560

General Index 853

every separable Hilbert space has a countable—, 562

examples, 560in L2[0, 2π ], 565–567is linearly dense, 560reordering, 823

orthonormal system, 459, 461, 560, 561, 563,796

output of a function, 3

PP, see set, all irrational numberspairwise nonoverlapping, 206parallelogram equality, 533, 551, 553, 556,

557, 571, 826Parseval identity, 562partial differential equation, 610, 829partial order, see order, partialpartial sum, 74, 660, 662, 696, 713

Abel’s formula, 81of a double series, 92of a Fourier series, 464, 465, 467, 472, 475,

476, 478, 482, 566, 568, 603, 607, 799of a rearrangement, 87, 664of a series of functions, 224of an alternating series, 84of the exponential series, 102of the function δ0, 607

partial sumof an unordered series, 863

partitionequally spaced —, 347finer than other, 202of a set, 2of an interval, 201tagged, 348, 745

finer than, 348partitioning, 645path, 540Peano axioms, 617Peano curve, 311perfect

set, see set, perfectsquare, 13

periodic distribution, 594, 596, 598characterization, 606convergence, 602convolution, 607, 827δ0, 599, 599derivative, 601, 601Fourier coefficients, 604Fourier series, 603function, 599, 600

order, 598permutation, 85point

accumulation, 44, 45, 71and limit of a function, 141in an infinite compact space, 50

at minimum distance, 556condensation, 288, 308extreme, 796, 817, 818, 824fixed, see fixed pointisolated, 44, 45, 56, 148, 288, 288, 308, 326,

331, 731, 737, 738pointwise

Cauchy, 216definition of operations, 189

pointwise convergenceof series, 224

pointwise limitof a function, 672

one-sided, 671, 748of a sequence, 215, 580of a sequence of continuous functions, 216of a sequence of functions, 221, 227, 338,

441, 705, 710–712, 753, 754Polish space, 306

decomposition perfect-countable, 308perfect, 308

cardinality c, 308only condensation points, 308

polynomial, 138, 239, 523, 720, 733, 810approximation by —, 221, 231, 233–235,

237, 305, 717coefficients, 138degree of a —, 138divisibility by monomial, 635, 758is continuous, 152is differentiable, 163, 173local approximation by —, 238, 244roots of a —,333, 681, 760, 807Taylor, see Taylor polynomialtrigonometric, 476, 688, 689, 769

approximation by —, 477, 482positional system, 4power

of cardinal numbers, 640of two sets, 640

power series, 250, 252–254, 256, 260, 269,716, 720

Abel’s criterion, 257coefficients of a —, 255, 257domain of convergence, 254, 257, 720infinitely differentiable, 255is analytic, 255, 260, 264

854 General Index

is infinitely differentiable, 251of binomial function, 280of the exponential function, 264of the logarithmic function, 268of the trigonometric functions, 274radius of convergence, 250, 252–255, 258,

260, 261, 264, 269, 274, 280, 716, 720then real analytic, 260

precompact, see totally boundedprime number, see number, primeprimitive function, see antiderivative, 797

calculus of —, 759, 761algebraic irrational, 761binomial, 764by parts, 756rational, 758transcendental, 764trigonometric, 769

probability density functionof a continuous random variable, 501

normal, 503of a discrete random variable, 490, 491

binomial, 499Poisson, 500two-point, 499

probability measure, see measure, probabilityprobability space, 488

discrete, 488finite, 488

property(D) of Darboux, 773Finite Intersection —, see Finite Intersection

PropertyHeine–Borel, 596, 596intermediate value, see Intermediate Value

PropertyLebesgue, see Lebesgue propertythree chord —, see Three Chord Property

Pythagorean triplet, 9, 632

Qquotient, 7

RR

complete, 20, 28, 29, 40, 41, 66, 73, 147,298, 311, 623, 624

connected, 52, 651Polish, 311separable, 304

Raabe test, see series, convergence criteria,Raabe

radius of convergence, see power series, radiusof convergence

radix point, 16, 21random variable, 489

discrete, 489mean, 493, 493, 494–499, 800median, 493mode, 493mutually independent —s, 492variance, 499vector-valued, 491

range of a function, 3, 135ratio test, see series, convergence criteria, ratio

testreal line, see R

real partof a complex functional, 548, 574of a complex number, 626

rearrangement of a series, 85, 85, 87, 664regular values, 576relatively compact set, see set, relatively

compactremainder

of a division, 7, 15, 635of Taylor polynomial, 243, 243

Cauchy form, 244Lagrange form, 244, 245, 250

of Taylor seriesCauchy form, 269Lagrange form, 259, 269, 277

resolvent set, 576restriction

of a function, 3Riemann function, 175, 219, 227, 414

does not have an antiderivative, 358has limit at each point, 673is Baire class 1, 219, 227is Riemann integrable, 355, 417points of continuity, 331

Riemann integral, 345implies Henstock–Kurzweil, 745implies Lebesgue, 417vector-valued, 772

Riemann lower sum, 343Riemann sum, 348

associated to a tagged partition, 745Riemann upper sum, 343root

Lemma, see lemma, RootNewton method for finding —, 249, 271of a �-system, 646of a polynomial, 333, 574, 635, 643, 681,

758–761, 807root test, see series, convergence criteria, root

test

General Index 855

Sσ -algebra, 117, 189, 191, 192, 390, 674

generated by a family, 122measure on a —, 121, 501, 503of all Borel sets, 122, 189, 195of all Lebesgue measurable sets, 118, 189,

195, 501scalar product, see inner productscattered space, 737Schauder basis, see basis, Schaudersecond category, see set, second categorysectio aurea, see golden ratiosegment, see initial segmentseparable, 46

Banach space, 532all homeomorphic, 522, 730c0 is —, 306c00 is —, 306C[0, 1] is —, 305compact metric is —, 313continuous image of a —, 304Hilbert space, 562

all linearly isometric, 562Lp is —, 306p is —, 305metric space, 302, 727, 736

characterization, 303, 322N

N is —, 309P is —, 304R is —, 46R is —, 304Rn is —, 304

T is —, 596totally bounded metric is —, 319

separation, 519completely separated, 676of points, 519, 542strong, 236

sequence, 5almost non-increasing, 658bounded, 62bounded above, 62bounded below, 62Cauchy, 72, 286decreasing, 66double, 91

iterared limit, 91increasing, 66limit, see limit, of a sequencenull, 286rapid decrease, 604slow growth, 603

strictly decreasing, 5, 6, 8, 66, 80, 100, 618,619, 702, 716

strictly increasing, 66, 100, 134, 618, 618,639, 735

sequence of functionsdecreasing, 222increasing, 222pointwise Cauchy, 216pointwise convergent, 215, 711uniformly bounded, 225uniformly convergent, 220, 222, 711

seriesalternating, 83alternating harmonic, 84associated to a sequence, 74bounded-multiplier convergent, 89Cauchy’s condensation criterion, 79convergence

absolute, 78necessary condition, 75subseries, 89unconditional, 85, 488unordered, 88

convergence criteriacomparison, 78, 79, 80, 86, 106, 247, 251,

659, 660, 663, 664Raabe, 80, 247, 248, 248, 661ratio test, 80, 80, 81, 253, 660root test, 80, 81, 661, 700

divergent, 74double, 92

absolutely convergent, 95comparison criterium, 93convergence, 96convergence criterium, 94necessary condition for convergence, 93Pringsheim criterium, 92reordering, 97Stolz criterium, 92summaton method, 94

geometric, 76of operators

absolutely convergent, 575power, see power seriesproduct, 97

Mertens criterium, 98sign-multiplier convergent, 90

set, 1all irrational numbers, 46, 138, 288, 309,

445, 583, 643, 711, 739, 746, 795cardinality, 36, 643dense, 37, 45homeomorphic to N

N, 310

856 General Index

in [0, 1], 125, 197, 443non complete, 73not compact, 315not complete, 298Polish, 309, 311residual, 329separable, 304, 305

all rational numbers, 288, 620, 789, 791dense, 37not complete, 298

all real numbers, see R

asymptotic, 613Bernstein, 646, 669, 669Borel, see Borel setbounded, 27bounded above, 27, 622bounded below, 27, 622closed, 39, 286compact, see compact setconnected, see connected setconvex, see convex, setcountable, 30countably infinite, 30dense, 288dense in R, 45derived, 288disconnected, 51, 52equivalent sets, 637finite, 3, 638, 639

then compact, 48first category, 54, 54, 327, 329–331, 652

and measure 1, 668Fσ , 113, 123, 207, 219, 412, 413, 651, 712,

725, 726not closed, 113Gδ , 217, 329, 651, 725, 726, 736, 741

and continuity, 217, 218, 674, 729and differentiability, 201, 588–591and extreme points, 818and measure, 114, 122, 123of a Polish space, 307

inevitable, see set, asymptoticinfinite, 638linearly dense, 519linearly ordered, see set, totally orderedmonotone functions

cardinality, 645not Gδ nor Fσ , 651nowhere dense, 54, 327null, 121of cardinality ℵ0, 30of cardinality c, 30

of cardinality of the continuum, see set ofcardinality c

open, 39, 53, 286open relatively to a set, 47orthonormal, 560partially ordered, 628perfect, 128, 128, 288, 308, 308porous, 787

implies nowhere dense, 787implies null, 787

preordered, 628relatively compact, 312, 321

characterization, 321, 740residual, 329, 330, 591, 668

characterization, 329second category, 54, 55, 327totally ordered, 628uncountable, 30well-ordered, 628

set of operatorsbounded, 526pointwise bounded, 526

setscompletely separated, 676similar, 646

similarity, 646singleton, 1singular part of a function, 428space

algebraic dual, see space, dual, algebraicBaire class 1

cardinality, 645Banach, see Banach spacebidual, 531, 823distortable, 615dual, 511, 511, 515, 517, 523, 539, 542, 590,

591, 596, 809, 816, 823, 827algebraic, 511, 515, 517topological, 511, 517, 598

Fréchet, 597hereditarily indecomposable, 614Hilbert, see Hilbert spacenon-degenerate, 536Polish

continuous image of the space of Baire,310

pre-Hilbertian, see inner product spaceRiemann integrable functions

cardinality, 644test functions for periodic distributions, 595topological vector —, 597

space of Baire, 309, 310, 583homeomorphic to P, 310, 311, 315

General Index 857

is Polish, 309spaces

linearly isometric, 511linearly isomorphic, 511

spectrum, 574, 576continuous —, 576point —, 576residual —, 577

standard deviation, 492of a random variable, 494

Stirling formula, 654, 654strictly decreasing

function, see function, strictly decreasingsequence, see sequence, strictly decreasingsequence of sets, 118

strictly increasingfunction, see function, strictly increasingsequence, see sequence, strictly increasingsequence of sets, 118

strongly exposed point, 823subalgebra, 236subbase, 624subcover, 47

finite, 47subdifferential, 444, 444, 449subfamily, 2, 47subseries, 89subset, 1

proper, 1subtangent, 444sum

of a series, 74of an arbitrary collection of nonnegative real

numbers, 646of cardinal numbers, 640of consecutive cubes, 632of consecutive squares, 632

superset, 1support

hyperplane, see hyperplane, supportof a function, 4, 294

support functional, see hyperplane, supportsupremum, 622, 651supremum of a bounded above set in R, 27system of generators of a vector space, 513

Ttags, 348tangent line, 161

equation, 163Taylor polynomial, 238, 240, 240, 241, 243,

245–247, 249, 255, 259, 277, 718, 719approximation by —, 242, 478

Taylor series, 258, 261, 721divergent, 259domain of convergence, 258not convergent to the function, 258of arcsin and arctan, 278of sin and cos, 277, 776, 778of a real analytic function, 260of logarithm, 268

telescopic argument, 354, 660theorem

Abel–Ruffini on the quintic, 760Alaoglu–Bourbaki, 522, 523, 549, 573, 584,

827Alexándrov, 306, 307, 308, 309, 311, 312,

329, 331, 736Arzelà–Ascoli, 332, 333, 364, 365, 730, 741Asplund–Lindenstrauss on differentiability

of convex functions, 590Auerbach on existence of Auerbach basis,

518, 543Baire Category

in R, 55, 55, 56, 217, 218, 328, 652, 741in metric spaces, 308, 328, 328, 329, 523,

545, 737, 738Baire on continuity of Baire class 1 functions,

217, 217, 218, 219, 711Baire’s Great, 219Banach Contraction Principle, 335, 336, 338,

367, 368, 370, 593, 680, 742Banach–Mazurkiewicz, 330Banach–Steinhaus, 482, 526, 545, 545, 550,

815Bauer maximum principle, 796Bishop–Phelps, 532Bolzano–Weierstrass, 71, 72, 73, 154, 320,

330, 349, 535, 655–658Borel on interpolation, 262Cantor–Bendixson, 308, 738Cantor–Bernstein–Schröder, 31, 32, 36, 638,

639, 641–644, 731Cauchy Mean Value, 169, 169, 174Cauchy–Peano existence, 363completeness of R, 28Darboux on the Intermediate Value Property

of the derivative, 167, 213, 358Dini on uniform convergence of a monotone

sequence, 222, 222, 223, 234, 716Dirichlet criterion on series, 82Dirichlet on fractions, 59Egorov on uniform convergence, 227, 228,

237, 372, 718Euclid on prime numbers, 9, 9, 632

858 General Index

Fermat on local extrema, 165, 165, 166–168,214, 586, 587, 692, 693

First Mean Value for Riemann integration,370, 371, 372, 420, 783

Fourier integral, 484Fubini on differentiation of series, 785Fundamental of Algebra, 574Fundamental of Arithmetic, 8Fundamental of Calculus

Lebesgue integral, 421–423, 423, 424, 425,427–429, 480, 503, 781, 789

Riemann integral, 356–358, 358, 359, 360,363, 368, 376, 379, 380, 427, 428, 595,750–753, 790

Heine–Borel, 45, 48, 50, 72, 153, 157, 319,596, 724

Heine–Cantor, 149, 157, 158, 181, 206,212, 290, 324, 324, 330, 332, 353, 676,678–680, 688, 702, 704, 706, 730

Helly’s first theorem, 700intermediate value property of continuous

functions, 155, 155, 156, 167, 214, 277,334, 371, 372, 675, 681, 695, 802

invariance of domain, 517James, 532Josefson–Nissenzweig, 811Krein–Milman, 519, 520, 796, 819Lagrange Mean Value, 166, 168, 168,

169–171, 210, 229, 256, 275, 359, 360,430, 447, 480, 612, 659, 678, 688, 696,698, 699, 704, 710, 714, 741, 742, 793

Lebesgue density, 786, 787Lebesgue Dominated Convergence, 91,

401, 401, 402–404, 406, 407, 425, 430,431, 433–435, 437, 503, 555, 701, 753,775–777, 785, 788

Lebesgue on convergence in measure, 237Lebesgue on differentiability of monotone

functions, 196, 198, 199, 201, 205, 210,361, 401, 422, 423, 428, 429, 591, 690,785

Lindenstrauss–Tzafriri on Hilbert spaces,615

Luzin, 193, 194, 667Mazur on differentiability of convex

functions, 588, 589Mean Value, see theorem, Lagrange Mean

ValueMinkowski–Carathéodory, 519, 521, 523,

524, 820Morse–Sard, 167, 207Nested Intervals, 40, 40, 41, 49, 50, 55, 84,

320, 651, 668, 747

Pettis measurability, 774Prime Number, 9Pythagoras, 554, 558, 559Ramsey, 645Riemann on rearrangements of series, 85–87,

87, 664Riemann–Lebesgue on Riemann integrabil-

ity, 347, 352–355, 380, 412, 413, 413,414–419, 746–748, 788

Riesz on the decomposition of a Hilbertspace, 556, 558, 559, 559, 560, 565, 573,574

Riesz on the dual of a Hilbert space, 570,571, 572, 573, 801, 811, 812, 816, 823

Root Lemma, 646Schauder on fixed points, 593Schur, 811, 827Second Mean Value for Riemann integration,

371, 383, 437, 473, 752Sierpinski on covering of R, 652Steinhaus, 668Steinhaus on the difference of two

measurable sets, 125, 125, 126, 131, 668Tietze, 194, 294, 294, 295, 296, 327, 676

for R, 676Tychonoff, 581, 582, 630Weierstrass approximation —, 138, 232,

235, 236, 305 478, 512, 689Zermelo, see Well-ordering principle

three chord property, 441topological space, 53topology, 53

of pointwise convergence, 578order —, 624weak, 583weak∗, 583w∗, see topology, weak∗

total variation, 202, 204totally bounded, 317

and complete equals compact, 320continuous image of a —, 322equals bounded on R

n, 319equivalent conditions, 318equivalent statements, 317implied by compact, 318implied by pseudocompact, 327implies bounded, 318implies separable, 319in complete spaces, 321non-compact, 322subspace of —, 318uniformly continuous image, 323

Transfinite Induction Principle, 648

General Index 859

translate of a set, 114triangle inequality, see inequality, triangletrigonometric system, 455, 566, 863

Uuniform convergence, see convergence,

uniformuniform limit

of a sequence of functions, 221, 222, 227,228, 298, 478, 688

of a sequence of Lipschitz functions, 733of a sequence of operators, 815of a sequence of polynomials, 231, 235, 689of a sequence of Riemann integrable

functions, 373uniformly equicontinuous, 324union of sets, 2unit interval, 14unit sphere, 507upper bound, 27, 622

for a preordered set, 628

Vvalue of a function, 3variance

of a random variable, 493vector space, 513

finite-dimensional, 513vectors

orthogonal, see orthogonal, vectorsVitali cover, 197Vitali nonmeasurable set, 131, 131

WWallis formula, 108, 784Wallis infinite product, 783Weierstrass M-test, 224, 231, 250, 251, 253,

257, 262, 281, 296, 700, 713, 714, 716Well-ordering principle, 629, 650

Symbol Index

AS ′, the set of all accumulation points of S, 288E′, the algebraic dual of the vector space E,

511(a.e.), almost everywhere, 121, 215ℵ0, the cardinal number of N, 638σm, the average of the first partial sums of the

Fourier series, 475

BB(X), the space of all bounded operators from

X into X, 510B(X,Y ), the space of all bounded operators

from X into Y , 510B[x0, r], the closed ball centered at x0 with

radius r , 286, 507BX , the closed unit ball of a normed space X,

507B(x0, r), the open ball centered at x0 with

radius r , 286, 507b, a base for a positional number system, 15a ∈ A, a belongs to A, 1X∗∗, the bidual space of X, 531bdr A, the boundary of a set A, 43

Cc, the cardinal number of R, 30, 638c0, the space of all null sequences, 507c00(�), the finitely supported vectors, 294C1P [0, 2π ], the space of continuously

differentiable 2π -periodic functions, 603C+, a Cantor ternary set of positive measure,

129cardA, the cardinal of a set A, 637A× B, the Cartesian product of A and B, 3λ, the complex conjugate of a complex number

λ, 551

�x�, the ceiling function, 26(CH), the Continuum Hypothesis, 638A, the closure of a set A, 43, 287T n, the n-times composition of an operator T

with itself, 575g ◦ f , the composition of two functions, 137Ac, the complement of a set, 1n = m (mod p), n congruent modulo p to m,

35CP [0, 2π ], the space of all continuous

2π -periodic functions, 567

DD, the dyadic numbers, 23(L,R), a Dedekind cut, 20(df/dx)x=a , the derivative of f at a, 160Dn, the n-th derivative operator, 595f ′(a), the derivative of f at a, 160f ′−(x), the left-hand-side derivative of f at x,

160, 199f ′+(x), the right-hand-side derivative of f at x,

160, 199diam (A), the diameter of a set A, 55, 286B \ A, difference of sets, 1dfa , the differential of f at a, 162D+f (x), Dini derivative of f at x, 198D−f (x), Dini derivative of f at x, 198D+f (x), Dini derivative of f at x, 198D−f (x), Dini derivative of f at x, 198δt , the Dirac delta at t 594, 598, 599, 602, 603Dhf (x0), the directional derivative of f in the

direction h at x0, 533F ⊕G, the direct sum of two spaces, 559dist (x,A), the distance from x to A, 289, 556Dm, the Dirichlet kernel, 465d(x, y), the distance between x and y, 283d2, the Euclidean distance, 283

© Springer International Publishing Switzerland 2015 861V. Montesinos et al., An Introduction to Modern Analysis,DOI 10.1007/978-3-319-12481-0

862 Symbol Index

d∞, the supremum distance, 285δij , the Kronecker delta, 513, 517D(f ), the domain of a function f , 3, 135dp , the distance in p , 293X∗, the dual space of X, 511

Ee, the base of the natural logarithms, 100∅, the empty set, 1A ∼ B, a set A is equivalent to a set B, 637ess sup (f ), the essential supremum of f , 294

Ff, the cardinality of the set of all real-valued

functions on R, 38, 644{Ai : ı ∈ I }, a family of sets, 2Pf (A), the family of all finite subsets of a set

A, 3A<N, the set of all finite sequences in A, 3�x�, the floor function, 26F ∼ ∑

n∈Zcnen, the Fourier series of a

distribution F , 603x ∼ ∑∞

i=1〈x, ei〉 ei , the Fourier seriesassociated to a vector x, 563

fr (x), the fractional part of x, 26(p, q), a fraction, 10f ′(x), the Fréchet derivative of f at x, 533

Xf−→ Y , a function from X into Y , 135

f : X→ Y , a function from X into Y , 135x $→ f (x), a function, 135

GGba := G(b) −G(a), 358gcd (a, b), the greatest common divisor of a

and b, 7graph f , the graph of a function f , 136

IIf , the imaginary part of a (complex) functional

f , 548⊂, the set inclusion, 627inf A, the infimum of a bounded below set in

R, 27〈·, ·〉, an inner product, 551[a, b], a bounded closed interval, 24Int A, the interior of a set A, 43⋂i∈I Ai , the intersection of a family of sets, 2∫Ef , the Lebesgue integral of f on E, 389

(a, b], a bounded left half open interval, 24(a, b), a bounded open interval, 24∫ baf , the Riemann integral of f on [a, b], 345

[a, b), a bounded right half open interval, 24f −1, the inverse of the function f , 137

f [n], the n-th iterated of f , 335

LL(X,Y ), the space of operators from X into Y ,

509L(I ), the set of all Lebesgue integrable

functions on I , 394λ, the Lebesgue measure, 121L1(E), the space of all classes of Lebesgue

integrable functions on E, 4111(N), the space of all absolutely summable

sequences, 5082(N), the space of all square-summable

sequences, 554λ∗, the Lebesgue outer measure, 110S, the Lebesgue singular function, 186lim inf, the limit inferior, 656lim infn xn, the limit inferior of a sequence

{xn}, 67f (x0−), the left-sided limit of f at x0, 143f (x0+), the right-sided limit of f at x0, 143lim sup, the limit superior, 656lim supn xn, the limit superior of a sequence

{xn}, 67λ∗, the Lebesgue inner measure, 124∞(�), the space of all bounded functions on

�, 285Lip(D), the set of all Lipschitz functions on D,

209ln x, the natural logarithmic function, 9, 273loga x, the basis-a logarithm, 273

MM, the class of Lebesgue measurable sets, 118UV , the set of mappings from V into U , 640E(X), the mean —or expected value — of a

random variable, 493(M , d), a metric space, 284−∞, minus infinity, 24

NN, the set of natural numbers, 4(n)b, a number n in base b, 15N

N, the space of Baire, 309||·||, a norm, 284, 506||·||∗, the dual norm, 511(XR, ‖·‖), the associated real normed space,

547‖ · ‖∞, the supremum norm, 285

OO(f ), Landau notation “big O”, 697o(f ), Landau notation “little o”, 697F⊥, the orthogonal to a set F , 553

Symbol Index 863

x ⊥ M , x orthogonal to a setM , 553x ⊥ y, x and y are orthogonal, 553ω(f , S), the oscillation of a function f on a set

S, 375ω(f , x), the oscillation of a function f at a

point x, 375

PP, the set of all irrational numbers, 22, 309, 643P / Q, partition P refinesQ, 343Q ≺ P , partition P refinesQ, 343P[a, b], the family of all partitions of the

interval [a, b], 343sF , a partial sum of an unordered series, 88PD, the space of periodic distributions, 598+∞, plus infinity, 24(f ∨ g), the pointwise maximum of two

functions, 235(f ∧ g), the pointwise minimum of two

functions, 235x−n, a negative power of x, 12uv, the power of two cardinal numbers, 640AB , the power set, 3xn, x to the power n, 11∑anx

n, a power series, 250f −1(U ), the preimage of a set U by a function

f , 150π (n), the number of prime numbers less than

or equal to n, 9∏∞n=1 an, an infinite product, 105

QQ, the set of all rational numbers, 10

RR, the set of all real numbers, 20R1(I ), the subset of L1(I ) of classes having a

Riemann integrable representative, 411R(f ), the range of a function f , 135Rf , the real part of a (complex) functional f ,

548ρ(T ), the resolvent of an operator T , 576f �A, the restriction of a function f to a set A,

3, 135R∗, the extended real number system, 24

Sf ′′(a), the second derivative of f at a, 164SF , the σ -algebra generated by a family F ,

122sign, the signum function, 142{x}, a singleton, 1σc(T ), the continuous spectrum of an operator

T , 576σp(T ), the point spectrum of an operator T ,

576σr (T ), the residual spectrum of an operator T ,

577σ (T ), the spectrum of an operator T , 576SX , the unit sphere of a normed space X, 507σ (X), the standard deviation of a random

variable, 4942V , the set of subsets of V , 640X(n), the set of all subsets of X of cardinality n,

645P(A), the set of all subsets of A, 638A ⊂ B, A is a subset of B, 12A, the family of all subsets of A, 2, 3P(A), the family of all subsets of A, 2∑γ∈� aγ , an unordered series, 88

supA, the supremum of a set A, 27supp (f ), the support of a function, 4B ⊃ A, B is a superset of A, 1

TT, the space of test functions for periodic

distributions, 595V ba f , the total variation of f on [a, b], 202CS, the trigonometric system in L[0, 2π ]459

U⊔, the disjoint union, 640⋃i∈I Ai , the union of a family of sets, 2

[0, 1], the unit interval, 14

VV (X), the variance of a random variable, 493

ZZ, the set of all integers, 6