# Compactness in $L(X, Y)$

Let $$X$$, $$Y$$ be Banach spaces and $$L(X, Y)$$ be the space of continuous linear mappings between them equipped with the operator norm. Is there a criterion for when a subset $$\mathcal{L} \subseteq L(X, Y)$$ is relatively compact? In my textbook, I could only find the well-known theorems about relative compactness in $$C(K)$$ and $$L^p(\mathbb{R})$$ for $$1 \leq p < \infty$$.

• I would be surprised if there were a useful criterion at that level of generality. Already when $X=Y=L^2(0,1)$ this seems pretty ambitious to me. Feb 11 at 21:16
• At the very least, when $X$ is locally compact (i.e. finite-dimensional), there is the usual criterion given by the Arzelà-Ascoli theorem. Feb 12 at 10:21
• @P.P.Tuong What form of the Arzelà-Ascoli theorem are you referring to? Feb 12 at 12:00
• @Smiley1000 A very general statement for various uniformities can be found in Bourbaki's General Topology, chap. X, § 2, no. 5. Note that since $L(X,Y)$ is Hausdorff and complete, relative compactness in $L(X,Y)$ of a subset of $L(X,Y)$ is equivalent to its precompactness for the induced unifomity Feb 12 at 13:25

It is possible only (as i see it) if you consider the operator $$T: L(X,Y) \to L(X,Y)$$ (say $$L$$) Banach spaces. $$T$$ here is a linear operator of operators. Then you can use: A linear operator is compact if and only if the image of any bounded set is relatively compact.
• But to decide whether $T$ is compact, we need to know what the compact subsets of $L(X, Y)$ look like, which is what this question is about. Feb 12 at 6:37