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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.
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