Remark

— 1) The topological vector space $\mathscr{S}(X, \mu; F)$ is not necessarily locally convex (Exer. 24). 2) The topology induced on the set of $f$ such that $N_p(f) \leq 1$ by the topology of convergence in measure may be strictly coarser than the topology induced on this set by the topology of convergence in mean of order $p$ (Exer. 22). However, see Prop. 21 below.