The compact is a property of space. In a nutshell, if space is compact, we can treat the space be a finite because space has a finite subcover. A continuous function on a compact space is uniformly continuous.
Heine-Borel theorem describes the condition of compactness of finite dimensional space. Closed and bounded But the Heine-Borel theorem does not hold in an infinite-dimensional space. We need another condition.
Previously, the compact space can be finite by taking subcover. The infinite-dimensional space can be finite by projection to finite dimension. If we could make as small as possible (i.e. \(\epsilon > 0\) ) the norm of \((X\backslash(1-P)\), the compactness is achieved.