The problem, then, is just to show that all parts of the definitions are satisfied.

"Linear Space". Show that both addition and scalar multiplication, f+ g and rf, where f and g are in C[a, b] and r is a number. Show that this has all the properties of a vectorspace:

Addition is associative: (f+ g)+ h= f+ (g+ h)

Addition is commutative: f+ g= g+ f

scalar multiplication distributes over addition: r(f+ g)= rf+ rg

r(sf)= (rs)f

There exist a "zero" function so that f+ 0= f for all f in C[a, b]

Every function has a "negative": for each f, there exist g such that f+ g= 0.

Norm: ||f||= sup |f(x)|.

Show that ||f|| is always non-negative and equal to 0 if and only if f is the "zero" function.

Show that ||rf||= |r| ||f|| where f is a function in C[a,b] and r is a number.

Show that .