For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.

By a *hyperplane* in a Banach space I understand any closed affine subspace of codimension 1.

By induction it can be shown that every Banach space $X$ of finite positive dimension has $\mathrm{cov}_H(X)=\mathfrak c$.

Problem 1.Is it true that $\mathrm{cov}_H(X)=\mathfrak c$ for any infinite-dimensional separable Banach space $X$?

In particular,

Problem 2.Is $\mathrm{cov}_H(\ell_2)=\mathfrak c$?

**Remark 1.** For any non-separable Hilbert space $X$ we have $\mathrm{cov}_H(X)=\omega_1$.