Pugh C.C. — Real Mathematical Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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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

equivalent      291
norm      285
$F_{\sigma}$ -set      189
$G_{\delta}$ -set      189
$r^{th}$ derivative      147
-area      315
$\alpha$ -Holder      186 253
$\delta$ -dense      253
$\epsilon$ , $\delta$ -condition      55
$\epsilon$ -chain      123
$\epsilon$ -principle      21
$\epsilon/2^n$ -argument      164
$\sigma$ -compact      250 257
1-form      314
Absolute continuity      398
Absolute convergence of a series      181
Absolute property      82
Abuse of notation      7
Accumulation points      69
Address string      97
Adheres      59
Advance map      234
Aleph null      30
Algebraic number      48
Almost everywhere      163
Almost uniform convergence      427
Alternating harmonic series      184
Alternating series      184
Ambient homeomorphism      105
Ambiently diffeomorphic      357
Analogy      10
Analytic function      148 235
Analyticity Theorem      237
Antiderivative      173
Antiderivative Theorem      173
Antoine’s necklace      107
ARC      122
Archimedean property      20
Area      306 364
Argument by contradiction      8
Arzela — Ascoli Theorems      214 216
Ascending k-tuple index      317
Ascending presentation      317
Associativity      14
Average (of a function)      396
Average derivative      278
Baire class 1      189
Baire’s theorem      243
Baker’s transformation      I 18
Banach Contraction Principle      228
Banach indicatrix      427
Banach space      285
Basic k-form      315
Bernstein polynomial      218
Bijection      30
Bilinear      275
Block test      197
Bolzano — Weierstrass theorem      77
Borel’s Lemma      256
Boundary of a k-cell      325
Boundary of a set      66 127
Bounded function      249
Bounded linear transformation      268
Bounded metric      124
Bounded sequence      49
Bounded variation      406 426
box      24
Brouwer’s fixed point theorem      228 334
Bump function      188
Burkill, J.C.      376
Cantor function      175
Cantor piece      103
Cantor set, fat      98 193
Cantor set, middle quarters      192
Cantor set, middle-thirds      95
Cantor space      103
Cantor surjection theorem      99
Cardinality      28
Cartesian product      21
Cauchy completion      112
Cauchy condition      18 73
Cauchy convergence criteria      19 180
Cauchy product of series      199
Cauchy sequence in a metric space      73
Cauchy — Binet formula      323 343
Cauchy — Riemann equations      340
Cauchy — Schwarz inequality      22
Cavalieri’s principle      305 338
Center of a starlike set      121
Chain connected      252
Chain rule      274
Chain-connected      123
Change of variables formula      306
Characteristic function      160
Chebyshev Lemma      401
class      2
Class , , , $C^{\infty}$ , $C^{\omega}$       147
Clopen      60
Closed form      329
Closed neighborhood      94
Closed set      59
Closed set condition      65
Closure of a set      66
Cluster point      69 126
Co-Cauchy      109
Coarse partition      155
Cohomology classes      333
Common refinement      158
Commutative diagram      291
Compact (in the covering sense)      88
Compact (sequentially)      76
Comparability of metrics      71
Comparable norms      346
Comparison Test      180
Complement of a set      41
complete      13 18
Completeness of a metric space      74
Completion Theorem      108
Complex analytic      238
complex derivative      339
Complex differentiable      239
Complex linear map      339
Composite      30
Compound word (as an address)      100
Concentration      395
Condensation point      69 126
Condition number      341
Conditional convergence of a series      181
Cone over a set      125
Connected      83
Connected component      136
Conorm      345
Constant rank      290
Content      419
Continuity      36 55 65
Continuously differentiable      147
Continuum Hypothesis      30
Contraction      228
Convergence of a sequence      17 54
Convergence with respect to an order relation      191
Convex      25
Convex combination      26 45
Convex function      46
Convex hull      105
Countable      30
Countable additivity      364 368
Countable base of a topology      128
Covering      88
Covering compact      88
Covering, Vitali or fine      392

Critical point      193
Critical value      193
Cupcake theorem      134
Curl of a vector field      329
Cut      11
Darboux continuous      144
Darboux integral      156
De Rham cohomology group      333
Decimal expansion      42
Decreasing sequence      117
Dedekind cuts      11
DeMorgan’s Law      41
Denjoy integral      423
Dense      98
density      395
Density point      395
Density, balanced      395 424
Denumerable      30
Derivate      403
Derivative      139 271
Derivative growth rate      235
Determinant      343
Devil’s ski slope      177
Devil’s staircase function      175
Diagonal      43
Diameter      79
Diffeomorphism      152 289
Difference of sets      2
Differentiable      141 271
Differential forms      313
Dipole      326
Directional derivative      348
Disconnected      83
Discontinuity of the first kind      194
Discontinuity of the second kind      194
Discrete metric      52
Disjoint sets      2
Distance from a point to a set      115
Distance vector      124
Divergence of a series      179
Divergence of a vector field      328
Domain of a function      28
Domination      180
Dot product      22
Double density point      426
Double integral      390
Dyadic      387
Dyadic filtration lemma      100
Dyadic number      44
Dyadic ruler function      193
Efficient covering      392
Egoroff ’s Theorem      427
Embeds      81
Empty set      2
Engulf      257
Envelopes, upper and lower      380
Equal cardinality      30
Equicontinuity      213
Equivalence relation, classes      3
Error factor      275
Euclidean distance      24
Euler characteristic      45
Euler’s product formula      200
Exact form      329
Excludes      94
Exponential growth rate      183
Extends to (of a function)      118
Exterior derivative      321
F sigma set      373
Fat Cantor set      98 193
Fatou’s Lemma      382
Field      15
Fine partition      155
Finite additivity      369
Finite cardinality      30
Finite intersection property      120
Finite subadditivity      419
Finite word      48
First orthant      24
Fixed point      43 228 334
Flow      233
Flux of a vector field      329
Frechet derivative      272
Front inclusion k-cell      326
Fubini’s Theorem      304
Function      28
Function algebra      223
Functional      314
Fundamental theorem of calculus      171
Fundamental Theorem of Continuous Functions      39
G delta set      373
Gap interval      103
Gauge      423
General linear group      351
GENERIC      243
Geometric series      180
Gradient vector field      298
Grand intersection      120
Graph      43
grid      300
Growing steeple      204
Hahn — Mazurkiewicz Theorem      132
Hairy ball theorem      359
Harmonic series      180 184
Hausdorff metric      133 199 252
Hawaiian earring      53 122
Heine — Borel theorem      77
Heine — Borel Theorem in $C^0([a, b],\mathbb{R})$       217
Henstock      423
Higher order differentiability      147
Hilbertcube      131
Holder condition      186 253
Holomorphic      239
Homeomorphism      57
Hull      374 417
identity map      30
Identity Theorem      256
Image set of a function      28
Implicit function      286
Implicit function theorem      286
Improper Riemann integral      178
Inclusion cell      318
Increasing sequence      117
Indefinite Lebesgue integral      398
Indicator function      160
Infimum      17
Infinite (cardinality)      30
Infinite product      198
Infinitely differentiable      147
inheritance      52 67
Inheritance principle      68
Initial condition      230
Injection      29
Inner measure      364 374
Inner product      27
Inner product space      27
Integer lattice      24
Integers      1
Integral test      181
Integrally equivalent functions      194
Integration by parts      178
Integration by substitution      177
Integration operators      330
Interior of a set      66 127
Intermediate value property      38 144
Intermediate Value Theorem      38 83 84
interval      19
Intrinsic property      82
Inverse function theorem      152 289
Inverse image      64

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