Assume a<b.

denotes the set of all continuous functions

and that all continuous functions

are integrable. So we have a function

given by

.

(I have already proved and concluded that I is a bilinear form).

Here's the question:

**Is this bilinear form non-degenerate? Justify your answer** **by stating explicitly any results you use from calculus.**
My Attempt

The bilinear form I is non-degenerate

1. If I(f,g)=0

then g=0

2. If I(f,g)=0

then f=0

I personally think this bilinear form is non-degenerate. I will try to give a proof by contradiction. For a contradiction suppose

but

.

I'm not sure how to carry on and finish this proof from this point. I don't know how to use calculus here... I know that integrals represent areas under the graphs. And in this case we know know

.

And f is not defined, if we define f = sin(t) and a=-1, b=1 we get

But this is a counter example to what we are trying to prove!

I have to submit this tomorrow, so I appreciate it if anyone could please help me complete this proof...urgently