Results 1 to 3 of 3

Math Help - Example for best approximation theorem

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    195

    Example for best approximation theorem

    The best approximation theorem states that if (H,<,>) is an inner product space and M is a non-empty, complete, convex subset of H, then for every x in H there is a unique y in M such that d(x,M) = ||y-x||.

    I don't know any good examples of inner product spaces (especially not complete IPS) with a complete convex subset. Any ideas?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    675
    Thanks
    32

    Re: Example for best approximation theorem

    An example of non complete inner product space is given by H:=\left\{f\in\mathcal{C}^1{\left[0,1\right]}, f(0=f(1)=0\right\} with the inner product \langle f,g\rangle :=\int_0^1 f(t)g(t)dt+\int_0^1 f'(t)g'(t)dt.
    An example of complete non empty convex subset is given by a finite dimensional subspace (it's works in each inner product space).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    195

    Re: Example for best approximation theorem

    So in this case, any polynomial space, Pn, for n finite.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Taylor's Theorem & Approximation (Spivak)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 1st 2011, 09:51 AM
  2. Replies: 4
    Last Post: January 10th 2011, 08:51 AM
  3. Central Limit Theorem Approximation
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 12th 2010, 09:58 PM
  4. Approximation using Taylor's Theorem
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 16th 2010, 09:19 AM
  5. Normal distibution approximation with central limit theorem
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: December 16th 2009, 03:27 AM

Search Tags


/mathhelpforum @mathhelpforum