Sequential characterization of compact operators

Suppose $T$ compact and let $(x_k)$ be a bounded sequence in $X$. By [A4] the range $\{x_k\}$ is empty or contained in some ball $B(x_0,r)$; because $\mathrm{\parallel }{x}_k\mathrm{\parallel }\le \mathrm{\parallel }{x}_k-x_0\mathrm{\parallel }+\mathrm{\parallel }{x}_0\mathrm{\parallel }$ for every $k$, there is a real $R\ge 0$ with $\mathrm{\parallel }{x}_k\mathrm{\parallel }\le R$ for all $k$ (if the range is empty take $R=0$), and then $T{x}_k\in R\overline{T({\stackrel{‾}{B}}_X)}$ for every $k$, because $x_k\in R{\bar{B}}_X$ and $T$ is linear.

Conversely assume every bounded sequence in $X$ has a subsequence whose $T$-images converge, and put $C:=\overline{T({\stackrel{‾}{B}}_X)}$. Let $(y_k)$ be a sequence in $C$. For each $k$ the set $S_k:=\{x\in {\bar{B}}_X:\mathrm{\parallel }{y}_k-Tx\mathrm{\parallel }<1\mathrm{/}(k+1)\}$ is nonempty, because $y_k\in \overline{T({\stackrel{‾}{B}}_X)}$, the set $T({\bar{B}}_X)$ is nonempty, and [A5] then gives a point of $T({\bar{B}}_X)$ within $1\mathrm{/}(k+1)$ of $y_k$; selecting $x_k\in S_k$ for every $k$ is a countable selection from nonempty sets, which [A3] licenses.

By [A1] the set $\overline{T({\stackrel{‾}{B}}_X)}$ is compact, and multiplication by the scalar $R$ is continuous, so $R\overline{T({\stackrel{‾}{B}}_X)}$ is a compact subset of $Y$ (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism); by [A2] its metric subspace is sequentially compact, so the sequence $(T{x}_k)$ of [step 1.1] has a subsequence converging in $Y$.

The sequence $(x_k)$ of [step 1.2] is bounded, since its range lies in ${\bar{B}}_X$; by hypothesis some subsequence $(x_{k_j})$ has $T{x}_{k_j}\to y$ for some $y\in Y$, and since every $T{x}_{k_j}$ lies in $T({\bar{B}}_X)\subseteq C$ while $C$ is closed, [A5] gives $y\in C$.

Therefore compactness of $T$ implies the stated sequential property for every bounded sequence.

Along that subsequence $\mathrm{\parallel }{y}_{k_j}-y\mathrm{\parallel }\le \mathrm{\parallel }{y}_{k_j}-T{x}_{k_j}\mathrm{\parallel }+\mathrm{\parallel }T{x}_{k_j}-y\mathrm{\parallel }<1\mathrm{/}(k_j+1)+\mathrm{\parallel }T{x}_{k_j}-y\mathrm{\parallel }$, and both terms tend to $0$, so $y_{k_j}\to y$ with $y\in C$.

Thus every sequence in $C$ has a subsequence converging in $C$, that is, $C$ is sequentially compact; by [A2] $C$ is compact, and then $T$ is compact by [A1].