I am recently learning functional analysis and topology, and I am wondering whether there is some relationship between compactness of a set (or a topological space) and compactness of a linear operator.
The definitions of two kinds of compactness are listed as below:
Definition 1: A topological space $(X, \tau)$ is compact iff every open cover of $X$ has a finite subcover.
Definition 2: Let $X$, $Y$ be normed spaces and let $T \in L(X,Y)$. Then $T$ is compact iff for every bounded sequence $\{ x_n \}$ in $X$, $\{Tx_n\}$ has a convergent subsequence.
Hope someone can help me with this. Thanks!