[Math] finite dimensional range implies compact operator

Question

Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in X$ is a bounded sequence, there exist a subsequence $x_{n_k}$ such that $Tx_{n_k}$ is convergent.
I want to prove that if $T\colon X\to Y$ has finite dimensional range, then it's compact.

Answer 1

Here is a direct proof.

Proof. Since $T$ has finite rank, Im$T$ is a finite-dimensional normed space. Furthermore, for any bounded sequence $\{ x_n \}$ in $X$, the sequence $\{ T x_n \}$ is bounded in Im$T$, so by Bolzano-Weierstrass theorem this sequence must contain a convergent subsequence. Hence $T$ is compact. $\square$