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fec to f, i.e one car

fec to f, i.e one car

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 93E
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Specifically part b

Problem 2. Consider the space C(R, R) of continuous real-valued functions. (a) Given any ƒ € C(R,R), show that there exists a sequence of polynomials (Pn)n that converges pointwise to f, i.e. limn→∞ Pn(x) = f(x) for every x ₹ R. (b) Show that one can moreover choose the sequence (Pn)n in such a way that the convergence is uniform on any compact subset of R, i.e. for every KCR compact, Pn → ƒ uniformly on K. (Note: the sequence Pn does not depend on K!)
Transcribed Image Text:Problem 2. Consider the space C(R, R) of continuous real-valued functions. (a) Given any ƒ € C(R,R), show that there exists a sequence of polynomials (Pn)n that converges pointwise to f, i.e. limn→∞ Pn(x) = f(x) for every x ₹ R. (b) Show that one can moreover choose the sequence (Pn)n in such a way that the convergence is uniform on any compact subset of R, i.e. for every KCR compact, Pn → ƒ uniformly on K. (Note: the sequence Pn does not depend on K!)
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