Schauder's theorem on adjoints of compact operators

Remarks

For the forward direction, let $K$ be the closure of the image of the unit ball of $X$; it is a compact metric space. The functionals in the unit ball of $Y'$, restricted to $K$, form a bounded and equicontinuous family of continuous functions on $K$. By the Arzelà–Ascoli theorem, every sequence $(g_n)$ in this ball has a subsequence that converges uniformly on $K$, and then $(T' g_n)$ converges in $X'$. For the converse, apply the forward direction to $T'$: the bidual $T''$ is compact, and $T$ is its restriction to $X$ under the canonical isometric embeddings $X \subseteq X''$ and $Y \subseteq Y''$. In a Hilbert space $H$, the same theorem says that a bounded operator $T$ is compact if and only if its Hilbert space adjoint $T^*$ is compact.

Going deeper

For a student

The theorem lets one move compactness between a space and its dual. It is a key step in the Riesz–Schauder theory of compact operators: it shows that $T - \lambda$ and $T' - \lambda$ have kernels of the same finite dimension for $\lambda \ne 0$. It also shows that the compact operators on a Hilbert space form a closed ideal stable under the adjoint, which is why the quotient by them is a C*-algebra.