  Given a Banach space we consider the $\sigma$-ideal of all of its subsets
which are covered by countably many hyperplanes and investigate its standard
cardinal characteristics as the additivity, the covering number, the
uniformity, the cofinality. We determine their values for separable Banach
spaces, and approximate them for nonseparable Banach spaces. The remaining
questions reduce to deciding if the following can be proved in ZFC for every
nonseparable Banach space $X$:
  (1) $X$ can be covered by $\omega_1$-many of its hyperplanes;
  (2) All subsets of $X$ of cardinalities less than ${\rm cf}([{\rm
dens}(X)]^\omega)$ can be covered by countably many hyperplanes.
  We prove (1) and (2) for all Banach spaces in many well-investigated classes
and that they are consistent with any possible size of the continuum. (1) is
related to the problem whether every compact Hausdorff space which has small
diagonal is metrizable and (2) to large cardinals.
