This text offers upper-level undergraduates and graduate students a survey of practical elements of real function theory, general topology, and functional analysis. Beginning with a brief discussion of proof and definition by mathematical induction, it freely uses these notions and techniques. The maximality principle is introduced early but used sparingly; an appendix provides a more thorough treatment. The notion of convergence is stated in basic form and presented initially in a general setting. The Lebesgue-Stieltjes integral is introduced in terms of the ideas of Daniell, measure-theoretic considerations playing only a secondary part. The final chapter, on function spaces and harmonic analysis, is deliberately accelerated. Helpful exercises appear throughout the text. 1959 edition.
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absolutely continuous Baire function basic neighborhood belongs Bernstein polynomial bounded variation Cauchy characteristic function closed interval complete contained continuous functions converges countably additive deﬁned deﬁnition denote derivative directed function elementary equality holds equation equivalent everywhere exercise exists f is continuous ﬁnd follows function f function of intervals functions on R4 Hausdorff space Hence hermitian induction inequality inﬁmum inﬁnite intervals of continuity L-functions Lebesgue measurable Lebesgue-Stieltjes integral LEMMA Let f Let g lim f lim inf lim sup limit linear functional linear space lower semicontinuous m-summable maximality principle me/ne measurable functions measurable sets metric space natural number non-negative nonempty norm notation open interval open set ordered ﬁeld orthogonal partially ordered set polynomial positive integer positive number PROOF prove real numbers real-valued functions Riemann-Stieltjes integrable satisﬁes step-function subdirected function subﬁeld summable function suppose supremum THEOREM topological space U-function uniformly continuous union upper bound