习题 5, § 7

令$(H_\lambda)_{\lambda \in L}$是向量空间$E$的超平面（经过到0）在域$K$上的族，它是$E$的覆盖。

(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) 如果$E$是有限维的，则证明$\operatorname{Card}(L) \geqslant 1 + \operatorname{Card}(K)$。 （通过归纳对$\dim(E)$进行争论。）