(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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]

2. Relevant equations

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]

3. 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.

Thanks!

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# Homework Help: Inner Product Space

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