Is the following an inner product space if the functions are real and their derivatives are continuous:
[tex] <f(t)|g(t)> = \int_0^1 f'(t)g'(t) + f(0)g(0) [/tex]
I was able to prove that it does satisfy the first 3 conditions of linearity and that
[tex] <f(t)|g(t)> = <g(t)|f(t)> [/tex]
But I was struggling with the last condition that:
[tex] <f(t)|f(t)> \geq 0 [/tex]
The Attempt at a Solution
I was able to get the following:
[tex] <f(t)|f(t)> = \int_0^1 [f'(t)]^2 + f(0)^2 [/tex]
Since this is square integrable, I figure that it must be greater than or equal to zero if the functions and their derivatives are continuous.
Am I right? It's the last term that's giving me some trouble.