Case
\begin{equation*}
A \in \mathscr{B}(X,Y) \, \text{compact} \Leftrightarrow A' \in \mathscr{B}(Y',X') \, \text{compact}.
\end{equation*}
where $X,Y$ are Banach spaces.
My proposal
Let $\eta_{n}$ be a sequence in $\mathscr{B}(X,Y)$ and $\mathbb{K} = \mathscr{B}'(Y',X')$. Consider the functions \begin{equation*} \psi_{n}(y) = \eta_{n}, \, y \in C(\mathbb{K}), \, \eta_{n} \in \mathscr{B}_{Y'}(X,Y). \end{equation*} The functions are equicontinuous and $\mathbb{K}$ is compact. So there is a converging subsequence $\psi_{n}$.
Comments
- I am not sure if the convergence has to be shown here.
- Taking sequences and applying them functions is intuitive but I would like to know better approaches and/or more rigorous if available.
Sources
- Compact operators, here
How can you prove the compactness of the adjoint Banach space with the compact origin space too?
