Inner Product Space Homework: Is <f(t)|g(t)> an Inner Product?

[NeedPhysHelp8]

Homework Statement

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

&lt;f(t)|g(t)&gt; = \int_0^1 f&#039;(t)g&#039;(t) + f(0)g(0)

Homework Equations

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

The Attempt at a Solution

I was able to get the following:
&lt;f(t)|f(t)&gt; = \int_0^1 [f&#039;(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]

I was able to get the following:
&lt;f(t)|f(t)&gt; = \int_0^1 [f&#039;(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!

Everything you have written down is non-negative. So, yes, you are correct.