Let $X$ and $Y$ be real Banach Spaces and let $$ K:X\rightarrow Y $$ be a bounded linear operator. Consider a sequence $\{u_k\}_{k=1}^\infty$ in $X$. We call a sequence $\{Ku_k\}_{k=1}^\infty$ to be precompact in $Y$ if there exists a subsequence $\{u_{k_j}\}_{j=1}^\infty$ such that $\{Ku_{k_j}\}_{j=1}^\infty$ converges in $Y$.
My question. What is the significance of the name, "precompact"? I can see its similarity with the definition of sequential compactness. However, I am not able to fully comprehend (appreciate) this similarity. Could you please give me more insights into this definition and the name, "precompact"? Put simply, why is "precompact" a good name? By "good," I mean a name that provides some information about the entity it defines.
A more general definition is that of precompact sets (as pointed out by hal4math). A subset $E$ of a topological space $X$ is said to be precompact if its closure is compact.
Many thanks!
In the books about Banach spaces, one would often see phrases like "a subset $A$ of a Banach space $X$ is relatively (weakly) compact", or "$A\subset X$ is (weakly) precompact.
$A\subset X$ is relatively compact if the norm closure $\overline{A}$ is compact. $A\subset X$ is precompact if every sequence in $A$ has a norm Cauchy subsequence. Since a Banach space is norm-complete, the concepts precompact and relatively compact coincide.
Similarly, $A\subset X$ is relatively weakly compact if the weak closure $\overline{A}^w$ is weakly compact. $A\subset X$ is weakly precompact if every sequence in $A$ has a weakly Cauchy subsequence. Every relatively weakly compact subset of $X$ is weakly precompact by Eberlein-Šmulian theorem. However, the converse is not generally true since (unlike the norm-topology) the weak topology is not sequentially complete for all Banach spaces.