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

Continuity of projection

  1. Dec 25, 2007 #1
    Are projections always continuous? If they are, is there simple way to prove it?

    If P:V->V is a projection, I can see that P(V) is a subspace, and restriction of P to this subspace is the identity, and it seems intuitively clear that vectors outside this subspace are always mapped to shorter ones, but I don't know how to prove it.

    If V was a Hilbert space, and we knew P(V) is closed, then I could prove this using the projection theorem. However only way to prove that P(V) is closed, that I know, is to use continuity of P.
  2. jcsd
  3. Dec 25, 2007 #2
    Is this a counter example?

    [tex]P:l^1\to l^1[/tex]

    [tex](1,0,0,0,0,0,0,\ldots)\mapsto (0,2,0,0,0,0,0,\ldots)[/tex]
    [tex](0,1,0,0,0,0,0,\ldots)\mapsto (0,1,0,0,0,0,0,\ldots)[/tex]
    [tex](0,0,1,0,0,0,0,\ldots)\mapsto (0,0,0,3,0,0,0,\ldots)[/tex]
    [tex](0,0,0,1,0,0,0,\ldots)\mapsto (0,0,0,1,0,0,0,\ldots)[/tex]
    [tex](0,0,0,0,1,0,0,\ldots)\mapsto (0,0,0,0,0,4,0,\ldots)[/tex]
    [tex](0,0,0,0,0,1,0,\ldots)\mapsto (0,0,0,0,0,1,0,\ldots)[/tex]

    hmh.. no it is not, because the mapping is not well defined, since (1,1,1,...) would be mapped to have infinite norm. But if we choose such vector spaces, where only finite amount of components can have non-zero values, then that could be it.

    (It seems I mixed [itex]l^1[/itex] and [itex]l^{\infty}[/itex].)
    Last edited: Dec 25, 2007
  4. Dec 25, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    just take a non closed subspace and a complement and project on the complement.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Continuity of projection
  1. Continuing series (Replies: 3)

  2. Continued fractions (Replies: 1)

  3. Continuity in algebra? (Replies: 18)

  4. Continued fractions (Replies: 3)