Exercise 5, § 7

Let $(H_\lambda)_{\lambda \in L}$ be a family of hyperplanes (passing through 0) of a vector space $E$ over a field $K$, which is a covering of $E$.

(a) Show that, if $K$ is finite, then $\operatorname{Card}(L) \geqslant 1 + \operatorname{Card}(K)$ and, if $K$ is infinite, $\operatorname{Card}(L) \geqslant \aleph_0 = \operatorname{Card}(\mathbf{N})$. (Prove by induction on $r$ that if $K$ has at least $r$ elements, then $E$ cannot be the union of $r$ hyperplanes.) Show by examples that these inequalities cannot be improved without an additional hypothesis (when $K$ is infinite, consider the space $E = K_s^{(N)}$).

(b) If $E$ is finite-dimensional, show that $\operatorname{Card}(L) \geqslant 1 + \operatorname{Card}(K)$. (Argue by induction on $\dim(E)$.)