Inner Product Space

  • #1

Homework Statement



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]


Homework 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]


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!
 

Answers and Replies

  • #2
LCKurtz
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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!
Everything you have written down is non-negative. So, yes, you are correct.
 

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