Let X be any metric space. Let $\mathcal{M}_X$ denote the class of all bounded subsets of a metric space $X$.
Definition:
- Let $(X,d)$ be a complete metric space. The function $\alpha:\mathcal{M}_X\rightarrow[0,\infty)$ with $(k = 1, 2, . . . , n\in\mathbb{N})$, s.t. \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \alpha(Q) &=& \inf\{\epsilon > 0: Q\subset \bigcup^n_{k=1}S_k, S_k\subseteq X, diam(S_k) <\epsilon\}\\ &=& \inf\{\epsilon>0: \text{$Q$ may be covered by finitely many sets of diameter }\leq\epsilon\} \end{eqnarray*} is called the Kuratowski measure of non-compactness.
Question: If X is infinite dimensional, how to show that $$\alpha(B_1(X))=\alpha(S_1(X))=2$$ where $B_1(X)$ and $S_1(X)$ are the unit ball and unit sphere in X , respectively.