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Exercise 7. Let X be a set. We denote by B(X, K) the set of functions f : X → K such that ||f||∞ < +∞, where ||f||∞ = sup|f(x)|. TEX

Exercise 7. Let X be a set. We denote by B(X, K) the set of functions f : X → K such that ||f||∞ < +∞, where ||f||∞ = sup|f(x)|. TEX

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 7. Let X be a set. We denote by B(X, K) the set of functions f : X → K such that ||f||∞ < +∞, where ||f||∞ = sup|f(x)|. TEX 1 1) Show that the set B(X, K) is a vector space for the usual addition of K-valued functions and the usual product of a scalar in K with a K-valued function. 2) Show that ||-||∞ is a norm on this vector space.
Transcribed Image Text:Exercise 7. Let X be a set. We denote by B(X, K) the set of functions f : X → K such that ||f||∞ < +∞, where ||f||∞ = sup|f(x)|. TEX 1 1) Show that the set B(X, K) is a vector space for the usual addition of K-valued functions and the usual product of a scalar in K with a K-valued function. 2) Show that ||-||∞ is a norm on this vector space.
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