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

Restrictions of compact operators

  1. Aug 10, 2011 #1
    Is it true that if [itex]T: X\to Y[/itex] is a compact linear operator, [itex]X[/itex] and [itex]Y[/itex] are normed spaces, and [itex]N[/itex] is a subspace, then [itex]T|_N[/itex] (the restriction of [itex]T[/itex] to [itex]N[/itex]) is compact? It seems like it would work, since if [itex]B[/itex] is a bounded subset of [itex]N[/itex], it's also a bounded subset of [itex]X[/itex] and hence its image is precompact in [itex]Y[/itex].

    But what if [itex]N[/itex] is just an arbitrary subset of [itex]X[/itex]? I guess it doesn't work in that case, though, since it doesn't even make sense to talk about [itex]T[/itex] being a linear operator in that case.
     
  2. jcsd
  3. Aug 10, 2011 #2
    Yes,the restriction will be compact by definition. When N is arbitrary, T would still make sense on the closed linear span of N.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook