Real Analysis

Cover, 1997 - 713 Seiten
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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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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...

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