for closed bounded sets in Rm , 49

Conway, 449

Dieudonn´, viii, 25, 60, 154, 206

e for metric spaces, 272“274

in Rm , 47

Halmos, 375, 391

Hardy, viii, 43, 83, 103, 154, 297 bounded variation, functions of, 181, 516“

Klein, 113, 422 519

Kline, 375 brachistochrone problem, 190

Littlewood, 81

Petard, H., 16 calculus of variations

Plato, 28 problems, 198“202

Poincar´, 376

e successes, 190“198

axiom used, 258

fundamental, 9, 12, 22, 374 Cantor set, 552

of Archimedes, 10, 12, 373, 412 Cauchy

of choice, 172, 252 condensation test, 76

axioms father of modern analysis, viii

for an ordered ¬eld, 379 function not given by Taylor series,

143

general discussion, 242, 364“365, 375“

377 mean value theorem, 457

Zermelo-Fraenkel, 375 proof of binomial theorem, 562

sequence, 67, 263

balls solution of di¬erential equations, 563

open and closed, 50, 245 Cauchy-Riemann equations, 479

packing, 233“235 Cauchy-Schwarz inequality, 44

packing in Fn , 237“238 Cayley-Hamilton theorem, 588

2

Banach, 242, 303 chain rule

Bernstein polynomial, 542 many dimensional, 131“132

607

608 A COMPANION TO ANALYSIS

one dimensional, 102“104 convergence tests for sums

Chebychev, see Tchebychev Abel™s, 79

chords, 475 alternating series, 78

closed bounded sets in Rm Cauchy condensation, 76

and Bolzano-Weierstrass, 49 comparison, 70

and continuous functions, 56“58, 66 discussion of, 465

compact, 421 integral comparison, 208

nested, 59 ratio, 76

closed sets convergence, pointwise and uniform, 280

complement of open sets, 51, 245 convex

de¬nition for Rm , 49 function, 451, 498“499

de¬nition for metric space, 244 set, 447, 571

key properties, 52, 245 convolution, 565“566

closure of a set, 526 countability, 383“386

comma notation, 126, 148 critical points, see also maxima and min-

compactness, 421, 536 ima, 154“160, 163“167, 340

completeness

D notation, 126, 150

de¬nition, 263

Darboux, theorems of, 452, 492

proving completeness, 267

decimal expansion, 13

proving incompleteness, 264

delta function, 221, 320

completion

dense sets as skeletons, 12, 356, 543

discussion, 355“358

derivative

existence, 362“364, 596

complex, 288“289

ordered ¬elds, 411“413

directional, 125

structure carries over, 358“361

general discussion, 121“127

unique, 356“358

in applied mathematics, 152, 401“

constant value theorem

404

false for rationals, 2

in many dimensions, 124

many dimensional, 138

in one dimension, 18

true for reals, 20

left and right, 424

construction of

C from R, 367“368 more general, 253

Q from Z, 366“367 not continuous, 452

R from Q, 369“374 partial, 126

Z from N, 366 devil™s staircase, 552

diagrams, use of, 98

continued fractions, 436“440

di¬erential equations

continuity, see also uniform continuity

and Green™s functions, 318“326

discussed, 7, 388“391, 417

and power series, 294, 563

of linear maps, 128, 250“253

Euler™s method, 577“580

pointwise, 7, 53, 245

existence and uniqueness of solutions,

via open sets, 54, 246

305“318

continuous functions

di¬erentiation

exotic, 2, 549“554

Fourier series, 302

integration of, 182“186

on closed bounded sets in Rm , 56“59 power series, 291, 557“558

term by term, 291

continuum, models for, see also reals and

under the integral

rationals, 25“28, 418“419

¬nite range, 191

contraction mapping, 303“305, 307, 330,

408 in¬nite range, 287

609

Please send corrections however trivial to twk@dpmms.cam.ac.uk

Dini™s theorem, 542 Greek rigour, 29, 365, 376, 521

directed set, 396 Green™s functions, 318“326, 583“586

dissection, 172

Hahn-Banach for Rn , 447

dominated convergence

Hausdor¬ metric, 534

for some integrals, 547

for sums, 84 Heine-Borel theorem, 449

duck, tests for, 369 Hessian, 157

hill and dale theorem, 164“166

economics, fundamental problem of, 58 H¨lder™s inequality, 531, 533

o

escape to in¬nity, 84, 283“284 homeomorphism, 600

Euclidean homogeneous function, 480

geometry, 364“365

norm, 44 implicit function theorem

Euler discussion, 339“347

method for di¬erential equations, 577“ statement and proof, 343“344

580 indices, see powers

on homogeneous functions, 480 inequality

Euler™s γ, 467 arithmetic-geometric, 451

Euler-Lagrange equation, 194 Cauchy-Schwarz, 44

exponential function, 91“98, 143, 317, 417, H¨lder™s, 531, 533

o

497, 591 Jensen™s, 450, 498

extreme points, 447“448 Ptolomey™s, 443

reverse H¨lder, 532

o

Father Christmas, 172

Tchebychev, 222

¬xed point theorems, 17, 303“304

in¬mum, 34

Fourier series, 298“302

in¬nite

Fubini™s theorem

products, 472, 561

for in¬nite integrals, 512

sums, see sums

for integrals of cts fns, 213, 510

injective function, 475

for sums, 90

inner product

full rank, 345

completion, 360, 364

functional equations, 477“479

for l2 , 531

fundamental axiom, 9

for Rn , 43

fundamental theorem of algebra

integral kernel, example of, 326

proof, 114“117

integral mean value theorem, 490

statement, 113

integrals

theorem of analysis, 114, 120

along curves, 228“229, 231

fundamental theorem of the calculus

and uniform convergence, 282

discussion of extensions, 186

improper (or in¬nite), 207“211

in one dimension, 184“186

of continuous functions, 182“186

over area, 212“217

Gabriel™s horn, 521

principle value, 211

Gaussian quadrature, 544

Riemann, de¬nition, 172“174

general principle

Riemann, problems, 205“206, 214

of convergence, 68, 263, 412

Riemann, properties, 174“181

of uniform convergence, 280

Riemann-Stieltjes, 217“224, 519

generic, 164

Riemann-Stieltjes, problems, 220

geodesics, 254“260

vector-valued, 202“204

global and local, contrasted, 65, 123, 142“

144, 155, 160, 164, 314“317, 341 integration

610 A COMPANION TO ANALYSIS

by parts, 189 method, 350“351

by substitution, 187 necessity, 350

numerical, 495“497, 544 su¬ciency, 353

Riemann versus Lebesgue, 206“207 Leader, examples, 528

term by term, 287 left and right derivative, 424

interchange of limits Legendre polynomials, 544“545, 559

derivative and in¬nite integral, 287 Leibniz rule, 580

derivative and integral, 191 limits

general discussion, 83“84 general view of, 395“400

in¬nite integrals, 512 in metric spaces, 243

integral and sum, 287 in normed spaces, 244

integrals, 213 more general than sequences, 55“56

limit and derivative, 285, 286 pointwise, 280

sequences in Rm , 46

limit and integral, 282“284

limit and sum, 84 sequences in ordered ¬elds, 3“7

partial derivatives, 149 uniform, 280

sums, 90 limsup and liminf, 39

interior of a set, 526 lion hunting

intermediate value theorem in C, 42

equivalent to fundamental axiom, 22 in R, 15“16, 58, 491“492

false for rationals, 2 in Rm , 48

not available in constructive analy- Lipschitz

sis, 418 condition, 307

obvious?, 25“28 equivalence, 248

true for reals, 15 logarithm

international date line, 108 for (0, ∞), 104“106, 476, 497

inverse function theorem non-existence for C \ {0}, 108“109,

alternative approach, 407“410 315“317

gives implicit function theorem, 342 what preceded, 475

many dimensional, 337

one dimensional, 106, 402 Markov chains, 571“574

inverses in L(U, U ), 336, 587 maxima and minima, 58, 154“160, 194“

irrationality of 202, 347“354

e, 97 Maxwell

√ 467

γ?, hill and dale theorem, 164

2, 432 prefers coordinate free methods, 46,

irrelevant m, 269 121

isolated points, 263 mean value inequality

for complex di¬erentiation, 289

Jacobian

for reals, 18“20, 22, 36, 60

determinant, 406

many dimensional, 136“138

matrix, 127

mean value theorem

Jensen™s inequality, 450, 498

Cauchy™s, 457

discussion of, 60

Kant, 28, 364

fails in higher dimensions, 139

kindness to animals, 514

for higher derivatives, 455

Krein-Milman for Rn , 448

for integrals, 490

statement and proof, 60

Lagrangian

limitations, 353 metric

611

Please send corrections however trivial to twk@dpmms.cam.ac.uk

as measure of similarity, 278“279 partial derivatives

British railway non-stop, 243 and Jacobian matrix, 127

British railway stopping, 243 and possible di¬erentiability, 147, 161

complete, 263 de¬nition, 126

completion, 363 notation, 126, 148, 150, 401“404, 423

de¬nition, 242 symmetry of second, 149, 162

derived from norm, 242 partition, see dissection

discrete, 273 pass the parcel, 339

Hausdor¬, 534 piecewise de¬nitions, 425

Lipschitz equivalent, 248 placeholder, 241, 350, 422

totally bounded, 273 pointwise compared with uniform, 65, 280,

M¨bius transformation, 255“261

o 282

monotone convergence power series

for sums, 470 addition, 459

and di¬erential equations, 294, 563

neighbourhood, 50, 245 composition, 470

non-Euclidean geometry, 364“365 convergence, 71

norm di¬erentiation, 291, 557“558

all equivalent on Rn , 248 limitations, 143, 298

completion, 358, 364 many variable, 469

de¬nition, 241 multiplication, 94

Euclidean, 44 on circle of convergence, 71, 80

operator, 128, 253, 481 real, 293

sup, 276 uniform convergence, 290

uniform, 275, 277

uniqueness, 293

notation, see also spaces

powers

Dij g, 150

beat polynomials, 434

Dj g, 126

de¬nition of, 109“113, 294“296, 555“

and · , 241

557

ι, 588

primary schools, Hungarian, 384

x, y , 43

primes, in¬nitely many, Euler™s proof, 473

g,ij , 148

probability theory, 221“224, 240“241

g,j , 126

Ptolomey™s inequality, 443

z — , 119

x · y, 43

quantum mechanics, 27

non-uniform, 422“423

nowhere di¬erentiable continuous func-

radical reconstructions of analysis, 375,

tion, 549

415“419

radius of convergence, see also power se-

open problems, 80, 468

ries, 71, 78, 290, 460

open sets

rationals

can be closed, 273

countable, 385

complement of closed sets, 51, 245

dense in reals, 12

de¬nition for Rm , 50

not good for analysis, 1“3

de¬nition for metric space, 244

reals, see also continuum, models for

key properties, 51, 245

and fundamental axiom, 9

operator norm, 128, 253, 481

existence, 369“374

orthogonal polynomials, 542

uncountable, 17, 385, 445

parallelogram law, 594“596 uniqueness, 380“381

612 A COMPANION TO ANALYSIS

Riemann integral, see integral dominated convergence, 84

Riemann-Lebesgue lemma, 566 equivalent to sequences, 68, 287

Rolle™s theorem Fubini™s theorem, 90

examination of proof, 453 monotone convergence, 470

interesting use, 63“64 rearranged, 81, 86, 467

statement and proof, 61“63 sup norm, 276

Routh™s rule, 485 supremum

routine, 50 and fundamental axiom, 37

Russell™s paradox, 375 de¬nition, 32

existence, 33

saddle, 157 use, 34“37

sandwich lemma, 7 surjective function, 475

Schur complement, 485 symmetric

Schwarz, area counterexample, 229 linear map, 481

Shannon™s theorem, 236“241 matrix, diagonalisable, 446

Simpson™s rule, 496

singular points, see critical points Taylor series, see power series

slide rule, 111 Taylor theorems

solution of linear equations via best for examination, 189

Gauss-Siedel method, 590 Cauchy™s counterexample, 143

Jacobi method, 590 depend on fundamental axiom, 145

global in R, 142, 189, 455

sovereigns, golden, 81“82

in R, 141“145

space ¬lling curve, 550

spaces little practical use, 190, 297

local in R, 142

C([a, b]), 1 ), 266, 278

local in Rn , 150“151, 154

C([a, b]), 2 ), 267, 278

C([a, b]), ∞ ), 278 Tchebychev

C([a, b]), p ), 533 inequality, 222

c0 , 270 polynomials, 454“456

l1 , 267 spelling, 455

l2 , 531 term by term

l∞ , 270 di¬erentiation, 287, 291, 302

lp , 533 integration, 287

s00 , 265 Thor™s drinking horn, 522

L(U, U ), 587 Torricelli™s trumpet, 521

L(U, V ), 253 total boundedness, 273

spectral radius, 588“590 total variation, 517

squeeze lemma, 7 transcendentals, existence of

Stirling™s formula, simple versions, 209, Cantor™s proof, 385

238, 504 Liouville™s proof, 435

successive approximation, 329“331 Trapezium rule, 495

successive bisection, see lion hunting trigonometric functions, 98“102, 143, 318,

summation methods, 461“464 519“520

sums, see also power series, Fourier se- troublesome operations, 306

ries, term by term and conver-

uniform

gence tests

continuity, 65“66, 182, 275

absolute convergence, 69

conditionally convergent, 78 convergence, 280“288

convergence, 68 norm, 275

613

Please send corrections however trivial to twk@dpmms.cam.ac.uk

uniqueness

antiderivative, 20, 186

completions, 356“358

decimal expansion, 13

Fourier series, 299

limit, 4, 47, 244

power series, 293

reals, 380“381

solution of di¬erential equations, 305“

308

universal chord theorem, 441

variation of parameters, 583

Vieta™s formula for π, 475

Vitali™s paradox, 171

volume of an n-dimensional sphere, 233

Wallis

formula for π, 472

integrals of powers, 494