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Burn, 62
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

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