Thread: Need urgent help! Integration in C0[0,1]

Need urgent help with this problem.

1)Let f be a function in C0[0,1] ie the space of continuous functions defined on the unit interval. Suppose that the Integral from 0 to 1 of the product of f and g (ie, f(x)*g(x) integrated from 0 to 1), for any g in C0[0,1] always vanishes. Prove that F is the constant zero function on [0,1].

2) What if g is restricted to being of the form g(x)=x^n for some natural n?

Many Many Thanks!

Originally Posted by frederick111

Need urgent help with this problem.

1)Let f be a function in C0[0,1] ie the space of continuous functions defined on the unit interval. Suppose that the Integral from 0 to 1 of the product of f and g (ie, f(x)*g(x) integrated from 0 to 1), for any g in C0[0,1] always vanishes. Prove that F is the constant zero function on [0,1].

2) What if g is restricted to being of the form g(x)=x^n for some natural n?

Many Many Thanks!

1) Consider g(x) = f(x)

2) Use the fact that polynomials are dense in C0 to show that the Integral from 0 to 1 of the product of f and g (ie, f(x)*g(x) integrated from 0 to 1), for any g in C0[0,1] always vanishes, and just use the result from part 1).,