Restrictions of compact operators

[AxiomOfChoice]

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

But what if N is just an arbitrary subset of X? I guess it doesn't work in that case, though, since it doesn't even make sense to talk about T being a linear operator in that case.

[Eynstone]

Yes,the restriction will be compact by definition. When N is arbitrary, T would still make sense on the closed linear span of N.