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Pugh C.C. — Real Mathematical Analysis

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Название: Real Mathematical Analysis

Автор: Pugh C.C.

Аннотация:

In this new introduction to undergraduate real analysis the author takes a different approach from past presentations of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians such as Dieudonne, Littlewood, and Osserman. This book is based on the honors version of a course which the author has taught many times over the last 35 years at Berkeley. The book contains an excellent selection of more than 500 exercises.

Язык:

Рубрика: Математика/Анализ/Функциональный анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 437

Добавлена в каталог: 25.04.2008

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Предметный указатель

Irrational numbers      19
Isometry      116
Iterate (of a function)      407
Iterated integral      390
Jacobian      306
Jordan content      306 419
Jordan curve theorem      132
Jump discontinuity      46 194
k-cell      315
k-chain      325
k-form      315
Kernel      374 418
Kurzweil      423
Lagrange multiplier      296
Lattice      24
Least upper bound property      13
Lebesgue density theorem      395
Lebesgue dominated convergence theorem      381
Lebesgue integrable      377
Lebesgue integral      376
Lebesgue measure      367
Lebesgue number      89 166
Lebesgue outer measure      363
Lebesgue sums      383
Lebesgue’s Fundamental Theorem of Calculus      396
Lebesgue’s Last Therorem      401
Lebesgue’s Main Theorem      399
Left limit      46
Leibniz rule      274
Length      363
Length of a vector      22
Less than or equal      13
Letters in an address      100
LIMIT      17 54 58
Limit infimum      49
Limit point      58
Limit set      61
Limit supremum      49
Linear transformation      267
Lipschitz condition      142 170 195 230
Littlewood’s princinples      394
Locally path-connected      132
Logarithm function      174
Lower bound      43
Lower integral      156
Lower semicontinuity      420
Lower sum      156
Lusin’s theorem      428
L’Hospital’s Rule      143
M-test      207 211 241 285
Magnitude      16
Magnitude of a vector      22
Manhattan metric      72 124
Maximum stretch      268
McLeod      423
Meager      243
Mean value property      141
Mean value theorem      141 277 278
Measurability (of a function)      376
Measurability, pre-image      383
Measurable      367
Measurable set      367
Measure continuity      370
Mertens’ Theorem      199
Mesh of a partition      155
Metaphor      9
Metric, metric space      51
Middle quarters Cantor set      192
Middle-thirds Cantor set      95
Minimum stretch      345
Modulus of continuity      253
Monotone      117
Monotone Convergence Theorem      377
Monotone function      47
Monotone sequence      117
Monotonicity of outer measure      364
Moore — Kline Theorem      103
Most (in the Baire sense)      243
Name of a form      314
Natural numbers      1
Nearly      392 412
Nearly continuous      412 429
Nearly uniform continuity      428
Nearly uniform convergence      412 427
Neighborhood      62
Nested sequence      78
Net x-variation      314
Norm      27
Normed space      268
Nowhere dense      98
Nowhere differentiable functions      240
One-to-one      29
Onto      29
Open covering      88
Open map      118
Open set condition      65
Operator norm      268
Orbit      118 407
Ordered field      15
Ordinate set      377
Orthant      24
Orthogonal linear map      341
Oscillating discontinuity      194
Oscillation      165
Outer measure      363
Outer measure axioms      364
Outer measure, abstract      367
p-adic      130
P-series      182
Parallelogram law      50
Partial derivative      272
Partition      104
Partition of an index set      353
Partition pair      154
Path      86
Path-connected      86 132
Peano curve      102
Peano space      132
Perfect      93
Periodic      407
Permutation      342
Picard’s Theorem      231
Piece      99
Piecewise continuous function      160
Piecewise linear function      245
Planar outer measure      364
Poincare’s Lemma      330
Pointwise convergence      201
Pointwise equicontinuity      213 249
Pointwise limit      201
Polar form      341
Positive definite symmetric linear map      341
Positive definiteness of a metric      52
Power series      185 211
Power set      47
Pre-image      64
Prime number      6
Product matrix      270
Propagation      216
Proper subset      82
Pullback      322
Pushforward      322
Quasi-round      412
Rademacher’s Theorem      406
Radius of convergence      185
Range of a function      28
Rank      290
Rank theorem      292

Ratio Mean Value Theorem      143
Ratio Test      183
Rational cut      12
Rational numbers      2
Rational ruler function      161
Real number      12
Rear inclusion k-cell      326
Rearrangement of a sequence      118
Rearrangement of a series      198
Reduces to (of a covering)      88
Refinement of a partition      158
Refinement Principle      158
Regularity (of Lebesgue measure)      373
Regularity hierarchy      148 208
Relative measure      395
Representative of an equivalence class      3
Residual      243
Retraction      334
Riemann $\zeta$ -function      200
Riemann integrable      155 300
Riemann integral      155
Riemann measurable      306
Riemann sum      154 300
Riemann — Lebesgue theorem      163 302
Riemann’s Integrability Criterion      160
Right interval      403
Right limit      46
Right slope      403
Root test      183
Sawtooth function      240
Schroeder — Bernstein theorem      34
Scraps      88
Second derivative      280
Second-differentiable      280
Semicontinuity      420
Separable metric space      128
Separates points      223
Separation (disconnectedness)      83
SEQUENCE      17 52
Sequentially compact      76
Sigma algebra      368
Sign of a permutation      342
Signed commutativity      316
Simple function      386 421
Simple region      356
Simpler-form      315
Simply connected      330
singleton set      2
Slice      387
Sliding secant method      145
Slope over an interval      401
Smooth      147 284
Somewhere dense      98
Space filling      102
Staircase approximation      356
Starlike      121 332
Steeple functions      204
Steinhaus’ Theorem      429
Step function      160
Stokes’ Theorem      326 327
Stone — Weierstrass theorem      223
Strictly decreasing sequence      117
Strictly increasing sequence      117
Subadditivity, finite      419
Subcovering      88
Subfield      15
Sublinear      271
Subsequence      54
subspace      52
Sup norm      205
Supersolving      225
Supremum      17
Surjection      29
Symmetric difference      2
Symmetrization of abilinearmap      350
Symmetry of a metric      52
t-advance map      234
Tag      423
Tail of a series      180
Tame      106
Target of a function      28
Taxicab metric      72 124
Taylor polynomial      149
Taylor series      151 235
Taylor’s Theorem      150 238
Telescoping sum      207
Term by term differentiation      210
Term by term integration      208
Thick      243
Thin      243
Tiling a manifold with cells      327
Topological equivalence      66
Topological property      64
Topological space      60
Topologist’s sine circle      53 123
Topologist’s sine curve      87
topology      60
Total derivative      272
Total length      98 363
Total length of a covering by intervals      163
Total variation      406 426
Totally bounded      92
Totally disconnected      96
Trajectory      230
Transitive relation      191
Transitivity      15
Translation      15
Translation (by a function)      378
Trefoil knot      105 134
Triangle inequality      16
Trichotomy      15
Trigonometric polynomial      227
Truncate an address string      97
Type 1 region, type 2 region      356
Ultrametric      129
Uncountable      30
Undergraph      154 376
Undergraph, completed      380
Undergraph, total      382 421
Uniform convergence      284
Uniform continuity      48 82 118
Uniform convergence      201
Uniform equicontinuity      249
Uniform limit      202
Uniformly Cauchy      284
Unit ball, cube, sphere      24
Universal compact metric space      99
Upper bound      13
Upper integral      156
Upper semicontinuity      136 264 420
Upper sum      156
Utility problem      132
Vanishing at a point      223
Variation      406
Vitali covering      392
Volume multiplier      306
Weak contraction      254
Wedge product      318
Weierstrass approximation theorem      217
Weierstrass M-test      207
Weighting signed area      315
Wild      106
Word (as an address)      100
Zeno’s maze      53
Zeno’s staircase function      161
Zero locus      256 286
Zero set      98 163 302 365
Zero-th derivative      147

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