hmwk4 - MATH 202A HOMEWORK 4 This assignment is due Wed.,...

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Unformatted text preview: MATH 202A HOMEWORK 4 This assignment is due Wed., Sept. 26. It deals with equicontinuity in C ( X ); the concepts of base, subbase, and subspace for a topological space; and a study of the the lower limit topology on R . Let X be a compact metric space, and let C ( X ) denote the set of all continuous complex-valued functions on X with the supremum norm. It is a theorem that a continuous function on a compact set is uniformly continuous, and thus the elements of C ( X ) are each uniformly continuous. This means that if f ∈ C ( X ), then ∀ > 0, ∃ δ > 0 such that ∀ x 1 ,x 2 ∈ X , if d ( x 1 ,x 2 ) < δ then | f ( x 1 )- f ( x 2 ) | < . A subset E of C ( X ) is called equicontinuous or an equicontinuous family if ∀ > 0, ∃ δ > 0 such that ∀ x 1 ,x 2 ∈ X and ∀ f ∈ E , if d ( x 1 ,x 2 ) < δ then | f ( x 1 )- f ( x 2 ) | < . Arzela-Ascoli theorem . Let X be a compact metric space. A subset E of C ( X ) is compact if and only if it is closed, bounded, and equicontinuous. The first three problems are exercises in understanding equicontinuity. Problem 1 . (a) Show that the set of functions { sin kt | k ∈ N } is not an equicontin- uous family in C ([0 , 2 π ])....
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This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.

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hmwk4 - MATH 202A HOMEWORK 4 This assignment is due Wed.,...

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