# Inner Product Space

## Homework Statement

Is the following an inner product space if the functions are real and their derivatives are continuous:

$$<f(t)|g(t)> = \int_0^1 f'(t)g'(t) + f(0)g(0)$$

## Homework Equations

I was able to prove that it does satisfy the first 3 conditions of linearity and that
$$<f(t)|g(t)> = <g(t)|f(t)>$$
But I was struggling with the last condition that:
$$<f(t)|f(t)> \geq 0$$

## The Attempt at a Solution

I was able to get the following:
$$<f(t)|f(t)> = \int_0^1 [f'(t)]^2 + f(0)^2$$
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.
Thanks!

LCKurtz
$$<f(t)|f(t)> = \int_0^1 [f'(t)]^2 + f(0)^2$$