Answer to Question #259371 in Functional Analysis for Jayanng

Question

Show that the space B[a,b] of all bounded real value functions on the interval [a,b] is a linear space over a vector field R

Answer 1

Let X be a nonempty set. A real-valued function f : X → R is bounded if there exists M > 0 so that |f(x)| ≤ M, for all x ∈ X. The set of bounded real-valued functions on X is denoted by B(X).

Given f, g ∈ B(X), and a ∈ R, we define (f + g)(x) = f(x) + g(x), (af)(x) = af(x) (properties of linear space)