Compact Embedding

Assume \(X\), \(Y\) are Banach spaces with \(X\subset Y\). We call \(X\) compactly embedded in \(Y\) if the inclusion map

\begin{equation*} i\colon X \to Y\colon x\mapsto x \end{equation*}

is compact .

It is customary to write \(X\subset \subset Y\) if \(X\) is compactly embedded in \(Y\).