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.