Trouble Demonstrating Hilbert Space Inner Product Definition

mkmath

I have attached an image of part of a derivation presented in a course that I am taking in Financial Engineering.

I am having trouble understanding/proving the jump from $$\int_0^{T_i}g(u) du=T_i g(0)+\int_0^{T_*}(T_i-u)^+g'(u)du$$ to $$\int_0^{T_i}g(u) du=<g,h>_H$$

In the course presentation deck I see the following statement that I have been unable to prove.

One of my course tutors responded to my question with this hint/ answer to how I should interpret $$<g,h>_H=g(0)h(0)+\int_0^{T_*}g'(u)h'(u)du$$...

I posted a question to the Calculus forum (http://mathhelpforum.com/calculus/280897-inner-product-integration-parts.html) but the more I dig into this, the more advanced it seems to become.

HallsofIvy

MHF Helper
I don't have time to look at the whole thing but I notice you say "I see the following statement that I have been unable to prove." That next statement is NOT a theorem to be proved. It is just defining the inner product as that calculation. Do you mean proving that IS an inner product? If so, start by stating the definition of "inner product". What do you need to prove to show that is an inner product?

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