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Intro to Analysis in-class_Part_54

Intro to Analysis in-class_Part_54 - 7.3 Uniform...

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Unformatted text preview: 7.3 Uniform convergence 113 7.3.4 Spaces of functions The previous result may be rephrased as: C [ a,b ] is dense in R [ a,b ] in the k · k 1-norm. Is C [ a,b ] dense in R [ a,b ] in the k · k ∞-norm? No: consider the Heaviside function H ( x ) = , x < 1 , x ≥ . For any f ∈ C ( R ), sup | H ( x )- f ( x ) | ≥ 1 2 by the IVT, so the sup cannot be made less than ε for ε < 1 2 . This is one manifestation of the fact that the topologies associated with the k·k 1-norm and the k · k ∞-norm are different. Here is another: a sequence which converges with respect to the k · k 1-norm but not the k · k ∞-norm. Define f n ( x ) := 2 nx, x ∈ [0 , 1 2 n ] , 2- 2 nx, x ∈ [ 1 2 n , 1 n ] , , else . Then f n is a triangular “tent function” with a symmetric peak of height 1 over the point x = 1 2 n and support [0 , 1 n ]. { f n } converges to f ( x ) ≡ 0 in the k · k 1-norm (and also pointwise) but not in k · k...
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Intro to Analysis in-class_Part_54 - 7.3 Uniform...

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