Foundations of Modern Analysis
Measure and integration, metric spaces, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces — all in a single study. Only book of its kind. Unusual topics, detailed analyses. Problems. Excellent for first-year graduate students, almost any course on modern analysis. Preface. Bibliography. Index.
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1ndeed 1t follows 1t is easily a-algebra a-finite a.e. real-valued absolutely continuous algebra assertion ball Banach space belong bounded linear operator called Cauchy sequence closed linear subspace closed set compact metric space completes the proof continuous function converges in measure Corollary countable defined Definition Denote disjoint union eigenvalue elements equation exists a sequence finite measure finite number function f Hence Hilbert space Hint inequality integrable function integrable simple functions interval Lebesgue measure Lebesgue-integrable Lemma Let xn linear functional linear subspace linear vector space measurable function measurable set measure space monotone nonnegative normed linear space open set orthonormal outer measure positive integer Problem Prove real line real number satisfies self-adjoint operator sequence xn sequentially compact set function signed measure space and let space H Suppose theory topological space topology uniformly weakly convergent x e H