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!

Norms in Hilbert Spaces.

  1. Aug 16, 2009 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Prove that for q>=p and any f which is continuous in [a,b] then [tex]|| f ||_p<=c* || f ||_q[/tex], for some positive constant c.

    2. Relevant equations
    The norm is defined as: [tex]||f||_p=(\int_{a}^{b} f^p)^\frac{1}{p}[/tex].

    3. The attempt at a solution
    Well, I think that because f is continuous so are f^p and f^q are continuous and on a closed interval which means they get a maximum and a minimum in the interval which are both positive (cause if f were zero then the norm would be zero and the ineqaulity will be a triviality), so f^q>=M2, f^p<=M1, and we get that:
    [tex]||f||_p/||f||_q<=M1^{1/p}/M2^{1/q}(b-a)^{1/p-1/q}[/tex] which is a constant.

    QED, or not?

    Thanks in advance.
  2. jcsd
  3. Aug 16, 2009 #2
    Your norm is wrong. It should be

    \|f\|_p=\left(\int_{a}^{b} |f|^p\right)^\frac{1}{p}

    This means that you need to argue using |f|, not f. Otherwise, the argument is OK.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Norms Hilbert Spaces Date
Bounding p-norm expression Dec 3, 2017
Help with this notation -- some sort of norm? Mar 4, 2017
Norm inequality, find coefficients Sep 5, 2016
Norm in Hilbert space Oct 7, 2012
Hilbert schmidt norm Dec 3, 2008