Answer to Question #123332 in Linear Algebra for alish

Question

a) Let scalar be real numbers. Show that the set of real-valued, continuous functions on
the closed interval [0 ,1] forms a vector space.
b) Also show that those functions in part (a) for which all nth derivatives exist for

Answer 1

A continuous [0,1] is a function

f:[0,1]→R

which is continuous for every point in [0,1]

Since the sum and scalar multiples of continuous functions are also continuous (and addition is commutative) we have a vector space since there are additive inverses (f−f=0) and a distinguished 0 element.