Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Example of dense non surjective operator

  1. May 25, 2009 #1
    Hi, can anyone give me an example of a bounded operator on a Hilbert space that has dense range but is not surjective? (Preferably on a separable Hilbert space)

    Im pretty sure such an operator exists since the open mapping theorem requires surjectivity and not just dense range, but its just bothering me that I can't find an example.
     
  2. jcsd
  3. Feb 9, 2011 #2
    How about \ell_1 (all absolutely summable sequences) being mapped into L_2[0, 2\pi) ( all complex valued square integrable functions on [0, 2\pi) ) by

    (a_n) -> \sum a_n \exp( \sqrt{-1} n x)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Example of dense non surjective operator
Loading...