Function Sequences Summary

Real Analysis · Axiom Academy

SUMMARY Sequences and Series of Functions A section recap: how uniform convergence preserves analytic structure, how power and Fourier series build functions, and how the sup-norm makes C[a,b] complete. Uniform beats pointwise. Pointwise convergence alone can lose continuity, integrability, and differentiability; uniform convergence — a single that works for every x at once — keeps them. One clean test for uniformity: uniformly on D exactly when the sup-norm . The Weierstrass M -test turns this into a numerical series check. What it preserves: continuity transfers, and the limit and integral interchange. Differentiation needs the extra hypothesis that the derivative sequence converges uniformly. Series machinery: a power series converges uniformly inside its radius R (Cauchy–Hadamard) and may be differentiated and integrated term-by-term there; Fourier series decompose periodic functions, with Parseval linking energy to coefficients. Function spaces: under the sup-norm, C[a,b] is complete — Cauchy-in-sup-norm sequences converge uniformly to a continuous limit — and Stone–Weierstrass says polynomials are dense. Core Concept Pointwise Convergence For each fixed x , the numbers f_n(x) converge to f(x) . The threshold N is allowed to depend on both and the point x — convergence can be arbitrarily slower at some points than others. Strength: easy to check one point at a time. Weakness: may fail to preserve continuity, , or . Core Concept Uniform Convergence

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