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

Elementary Linear Algebra (MindTap Course List)
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Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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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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