# Real Analysis

ClassicalRealAnalysis.com, 1997 - 713 Seiten
This book provides an introductory chapter containing background material as well as a mini-overview of much of the course, making the book accessible to readers with varied backgrounds. It uses a wealth of examples to introduce topics and to illustrate important concepts. KEY TOPICS: Explains the ideas behind developments and proofs — showing that proofs come not from “magical methods” but from natural processes. Introduces concepts in stages, and features applications of abstract theorems to concrete settings — showing the power of an abstract approach in problem solving.

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### Inhalt

 Measure Spaces 63 Metric Outer Measures 111 Measurable Functions 160 Integration 187 Fubinis Theorem 247 Diﬀerentiation 263 Diﬀerentiation of Measures 308 Metric Spaces 347
 10Baire Category 406 Analytic Sets 447 Banach Spaces 476 The Lp spaces 535 Hilbert Spaces 574 Fourier Series 613 Index 659 Urheberrecht

### Beliebte Passagen

Seite 369 - X is called a Cauchy sequence if for every e. > 0 there exists an integer N such that d(.\q, xm) < t: whenever q, m > N.
Seite 409 - A set that is not of the first category is called a set of the second category. 3. The complement of a first-category set is called a residual set. For complete metric spaces, first-category sets are the "small" sets and residual sets are the "large" sets in the sense of category.
Seite 549 - The sum of the squares of the diagonals of a parallelogram equals the sum of the squares of its sides.
Seite 369 - A metric space is said to be complete if every Cauchy sequence in this space converges.
Seite 359 - T is continuous at x if and only if, for every e > 0, there is a <5 > 0 so that a(T(x), T(y)) < e, whenever p(x, y) < 5. Also T is continuous at every point in X if and only if, for every open set GCY, the set T~l(G) = {xeX : T(x) e G] is open.
Seite 181 - We are now ready to state and prove the main theorem of this paper. Theorem.
Seite 368 - Prove that a metric space X is separable if and only if there exists a countable collection U of open sets such that every open set in X can be expressed as a union of members of U.
Seite 385 - Show that if /:X— > Y is uniformly continuous and {xn} is a Cauchy sequence in X, then {/(*„)} is a Cauchy sequence in Y.
Seite 115 - A metric space (X, d) is said to be separable if there exists a countable subset of X that is dense in X.
Seite 220 - Riemann integrable over [a, b] if and only if for every e > 0, there exists...