Results 1 to 1 of 1

Math Help - [SOLVED] Orthonormal basis in Hilbert space

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    14

    Orthonormal basis in Hilbert space

    Hi,
    The first part of a problem was to show that if Y is a dense subspace of a separable hilbert space H, then H has an orthonormal basis consisting of elements in Y.
    I was able to do this.
    The second part of the problem is use this result to prove that if f\in L^1(\mathbb{R})\setminus L^2(\mathbb{R}), there is an orthonormal basis \{g_n\} of continuous bounded functions such that \int_{\mathbb{R}} f(x)g_n(x)\, dx = 0.

    I can see that if we define the linear functional T, with domain the set of continuous functions of compact support C_0(\mathbb{R}) intersected with L^2(\mathbb{R}), by T(g) = \int_{\mathbb{R}} f(x)g(x) \, dx, then we want to take Y to be \ker T and H = L^2(\mathbb{R}) in the above result. It suffices to show \ker T is dense in C_0(\mathbb{R})\cap L^2(\mathbb{R}) since C_0(\mathbb{R}) is dense in L^2(\mathbb{R}). I know that this is equivalent to T being discontinuous (ie. unbounded), but I'm getting stuck on using that f isn't in L^2(\mathbb{R}) to show T is unbounded.

    Is this the right idea? If so, how can we use that f is not in L^2(\mathbb{R}) to show that T is unbounded?

    Many thanks.

    Edit: I hadn't solved the problem, but thought the above was too wordy so wished to replace it.
    Last edited by james123; May 1st 2010 at 06:00 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basis on Hilbert space
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 5th 2012, 11:06 AM
  2. Orthonormal Sequence in a Hilbert space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: June 3rd 2011, 09:35 PM
  3. Orthonormal Basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 25th 2011, 06:00 AM
  4. Inverse of Mapping from Hilbert Space to Hilbert Space exists
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 2nd 2009, 08:15 PM
  5. Orthonormal basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 1st 2008, 05:19 AM

Search Tags


/mathhelpforum @mathhelpforum