Restrictions of compact operators

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SUMMARY

If T: X → Y is a compact linear operator between normed spaces X and Y, then the restriction of T to a subspace N, denoted T|_N, is indeed compact. This conclusion holds true as long as N is a closed linear subspace of X. If N is merely an arbitrary subset, the restriction does not maintain the properties of a linear operator, thus complicating the definition of compactness. Therefore, the compactness of T|_N is guaranteed only when N is a closed linear subspace.

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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.
 
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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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