1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Inner Product Space

  1. Oct 27, 2011 #1
    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.
  2. jcsd
  3. Oct 27, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Everything you have written down is non-negative. So, yes, you are correct.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook