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Thread: Trouble Demonstrating Hilbert Space Inner Product Definition

  1. #1
    Junior Member mkmath's Avatar
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    Trouble Demonstrating Hilbert Space Inner Product Definition

    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 $$\int_0^{T_i}g(u) du=<g,h>_H$$

    Trouble Demonstrating Hilbert Space Inner Product Definition-capture.jpg

    In the course presentation deck I see the following statement that I have been unable to prove.
    Trouble Demonstrating Hilbert Space Inner Product Definition-capture2.jpg

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

    Trouble Demonstrating Hilbert Space Inner Product Definition-capture3.jpg

    I posted a question to the Calculus forum (Inner product from integration by parts) but the more I dig into this, the more advanced it seems to become.
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  2. #2
    MHF Contributor

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    Re: Trouble Demonstrating Hilbert Space Inner Product Definition

    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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  3. #3
    Junior Member mkmath's Avatar
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    Re: Trouble Demonstrating Hilbert Space Inner Product Definition

    Oh! I never thought about it from that point of view...I assumed that I could derive it directly.

    Instead I should be taking the statement $$\int_0^{T_i}g(u)du=T_ig(0)+\int_0^{T_*}(T_i−u)^+ g′(u)du$$ as a definition and then show that it satisfies the properties of an inner product.

    I found this nice thread that helped:https://math.stackexchange.com/quest...-inner-product

    Thanks again HallsofIvy!
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