Foundations of Mathematical AnalysisCourier Corporation, 01/01/2002 - 429 من الصفحات This classroom-tested volume offers a definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. A self-contained text, it presents the necessary background on the limit concept. (The first seven chapters could constitute a one-semester course on introduction to limits.) Subsequent chapters discuss differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue integral. More than 750 exercises help reinforce the material. 1981 edition. 34 figures. |
المحتوى
Preface to the Dover Edition | 1 |
The Real Number System | 9 |
Set Equivalence | 26 |
Sequences of Real Numbers | 34 |
6 | 64 |
Infinite Series | 73 |
26 | 81 |
29 | 92 |
Measure Zero | 230 |
A Necessary and Sufficient Condition for the Existence of the Riemann Integral | 234 |
Improper RiemannStieltjes Integrals | 238 |
Sequences and Series of Functions | 245 |
Integration and Differentiation of Uniformly Convergent Sequences | 249 |
Series of Functions | 253 |
Applications to Power Series | 259 |
Abels Limit Theorems | 262 |
on the Real Line | 102 |
33 | 109 |
34 | 112 |
Metric Spaces | 116 |
38 | 128 |
40 | 136 |
45 | 155 |
48 | 171 |
MeanValue Theorems and LHospitals Rule | 176 |
Taylors Theorem | 185 |
The RiemannStieltjes Integral | 189 |
RiemannStieltjes Integration with Respect to an Increasing Integrator | 190 |
RiemannStieltjes Sums | 204 |
RiemannStieltjes Integration with Respect to an Arbitrary Integrator | 210 |
Functions of Bounded Variation | 213 |
RiemannStieltjes Integration with Respect to Functions of Bounded Variation | 219 |
The Riemann Integral | 225 |
Summability Methods and Tauberian Theorems | 265 |
Transcendental Functions | 268 |
The Natural Logarithm Function | 271 |
The Trigonometric Functions | 274 |
Inner Product Spaces and Fourier Series | 280 |
The Inner Product Space R³ | 285 |
Inner Product Spaces | 288 |
Orthogonal Sets in Inner Product Spaces | 293 |
Normed Linear Spaces and the Riesz Representation | 335 |
The Dual Space of a Normed Linear Space | 343 |
Proof of the Riesz Representation Theorem | 349 |
The Lebesgue Integral | 355 |
Vector Spaces | 405 |
Hints to Selected Exercises | 411 |
| 421 | |
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عبارات ومصطلحات مألوفة
a₁ a₂ am,n an+1 b₁ BV[a C₁ Cauchy sequence Cauchy-Schwarz inequality closed interval closed subset compact metric space complete metric space continuous function convergent subsequence converges absolutely converges uniformly Corollary countable d₁ d₁(X defined diverges equation Exercises exists a positive finite Fourier series function ƒ ƒ and g ƒ is bounded ƒ is continuous improper integral infinite series inner product space Lemma Let f Let ƒ lim inf lim sup limit point M₂ measurable functions metric space M₁ N₁ nonnegative normed linear space open ball open interval open set open subset partial sums partition positive integer power series proof of Theorem Prove that ƒ radius of convergence rational numbers real numbers real sequence Riemann-Stieltjes integral Section sequence of partial space and let sum by rows U₁ uniformly continuous vector space verify x₁