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.
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