Exercise 25, § 5

Let $E$ be an infinite dimensional Fréchet space of enumerable type (therefore having an enumerable everywhere dense subset) (cf. I, p. 25, exerc. 1). Show that there exists an everywhere dense hyperplane $H$ of $E$ which meets every closed, infinite dimensional linear variety of $E$. (Use the existence of a basis, having the cardinal of the continuum, in each of the direction subspaces of these varieties (exerc. 24, c)) and the fact that the set of closed, infinite dimensional, linear varieties of $E$ also has the cardinal of the continuum (GT, IX, § 5, exerc. 17); then apply a method of construction of a linear form on $E$ following from S, III, § 6, exerc. 24.) The hyperplane $H$ does not contain any infinite dimensional, closed, vector subspace.