A Course in Analysis:Volume I: Introductory Calculus, Analysis of Functions of One Real Variable: Volume 1 by Niels Jacob

Author:Niels Jacob [Jacob, Niels]
Language: eng
Format: azw3
Publisher: Wspc
Published: 2015-08-18T04:00:00+00:00

for 1 ≤ j ≤ n.

11. Let be a sequence converging in the norm || · ||p, p ∈ [1, ∞), to some x = (x(1), . . ., x(n)) ∈ n. Suppose that || · || is a further norm on n satisfying the inequality ||y|| ≤ c||y||p for all y ∈ n with some c > 0. Prove that converges to x with respect to || · ||.

24 Uniform Convergence and Interchanging Limits

A lot of the material in this chapter can be skipped during a first reading. Of importance are the definitions of pointwise and uniform convergence, the fact that uniform convergence can be described as convergence with respect to the supremum norm and the result that the uniform limit of continuous functions is continuous, Theorem 24.6. However here is the correct place to add some further material to be considered later.

In the following let K ≠ be a set. We may consider functions f, g : K → and for α ∈ it follows that the functions f ± g, f · g and αf can be defined on K by

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