Real AnalysisFor beginning graduate-level courses in Real Analysis, Measure Theory, Lebesque Integration, and Functional Analysis. An important new graduate text that motivates the reader by providing the historical evolution of modern analysis. Sensitive to the needs of students with varied backgrounds and objectives, this text presents the tools, methods and history of analysis. The text includes the topics that "every graduate student must know" as well as specialized topics that prepare students for further study in analysis. User-friendly in approach, it includes numerous examples and exercises to illustrate and motivate the concepts. |
Other editions - View all
Common terms and phrases
absolutely continuous algebra approximation arbitrary Baire Banach space Borel sets bounded variation Cantor set Chapter closed sets compact complete contains continuous function convergence countable definition denote derived number differentiable Dv(x eigenvalue element example Exercise exists f(xo fn(x follows Fourier series function defined function f ƒ is continuous Hilbert space Hint inequality infinite inner product integrable functions IRē Lebesgue integral Lebesgue measure Lebesgue Stieltjes measure Lemma Let ɛ Let f Let ƒ linear operator mapping measurable functions measurable sets measure space measure zero Method nondecreasing nonempty nonnegative normed linear space notion obtain open interval open sets pairwise disjoint pointwise polynomial proof of Theorem Prove real numbers Riemann integral Section separable metric space set function Show signed measure subset subspace Suppose theory trigonometric type Gs union verify Vitali cover
Popular passages
Page 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.
Page 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.
Page 549 - The sum of the squares of the diagonals of a parallelogram equals the sum of the squares of its sides.
Page 369 - A metric space is said to be complete if every Cauchy sequence in this space converges.
Page 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.
Page 181 - We are now ready to state and prove the main theorem of this paper. Theorem.
Page 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.
Page 385 - Show that if /:X— > Y is uniformly continuous and {xn} is a Cauchy sequence in X, then {/(*„)} is a Cauchy sequence in Y.
Page 115 - A metric space (X, d) is said to be separable if there exists a countable subset of X that is dense in X.
Page 220 - Riemann integrable over [a, b] if and only if for every e > 0, there exists...