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.