How to show that the following set is compact.

Question

Let $T : l_2 \to l_2$ be a continuous linear operator. Let $B$ be the closed unit ball in $l_2$. How to show that the closure of $T(B)$ i.e. $\overline{T(B)}$ compact in $l_2$ where $T(x) = (x_1, x_2, \ldots, x_{10}, 0, 0, \ldots)$, $x = (x_n) \in l_2$.

Answer 1

For all $x \in \ell_2$, you have $\Vert T(x) \Vert_2 \le \Vert x \Vert_2$. Hence $T(B)$ is bounded and also $\overline{T(B)}$ as the closure of a bounded set is bounded (by the same bound).

$T(B)$ is finite dimensional and its adherence also. Finally $\overline{T(B)}$ is a closed bounded set in a finite dimensional space. Therefore it is compact.

Answer 2

Note that $T(B)\subset\{(x_1,x_2,\ldots,x_{10},0,0,\ldots)\,|\,x_1,x_2,\ldots,x_{10}\in\mathbb{R}\}$. This set is basically $\mathbb{R}^{10}$, in the sense that it is isometric to it. Therefore, $\overline{T(B)}$ is compact, since it is closed and bounded.