continuously differentiable real-valued functions

[craig11]

Let H'[0,T]be the space of real-valued functions which are continuously differentiable on [0,T]. Show that <f,g> = integral from (0,1) [f(t)g(t)dt + f'(t)g'(t)] dt define an inner product on H'[0,T]

I am stumped on this question, need to know how to do it for a test in a day. any help would be greatly appreciated.

 

[pka]

Let H'[0,T]be the space of real-valued functions which are continuously differentiable on [0,T]. Show that <f,g> = integral from (0,1) [f(t)g(t)dt + f'(t)g'(t)] dt define an inner product on H'[0,T]

I will ask you to review the wording of what you posted!
Do you mean ‘T’ or ‘1’. It seems that you have confused the two.
Next question for you: “Do you know the definition of inter product”?
You can note that the product of two continuous functions is continuous.
Also, continuous functions are integrable; and you may want to use integration by parts.

 

[craig11]

i have rechecked the question and everything i have written is identical to the question.
i know an example of an inner product is the dot product. so product of 2 vectors giving u a number
should i be splitting up the 2 terms within the integration sign.. like make it so it's integral ____ + integral ______ instead of one big integral.
but i am not sure what i am looking for, even if i do take the integration of it.

 

[pka]

Sorry but vector products have nothing to do with this problem.
I do not see how to evaluate \(\displaystyle \int\limits_0^1 {fg + f'g'}\) in general.
In particular, if \(\displaystyle T = \frac{1}{2}\) I think there is a counter-example to this proposition.

 

[craig11]

how would you go about doing it if the question was in fact wrong ? apparantly this question was on last years test also so i think it is correct, but regardless, i don't even know how to start it