Corollary 2

— Given n points $a_k$ of F ($1 \leq k \leq n$), and n numerical functions $f_k$ ($1 \leq k \leq n$) belonging to $\mathscr{L}^p$, the function $f = \sum_{k=1}^n a_k f_k$ belongs to $\mathscr{L}_F^p$.

This follows from the fact that the mapping $t \mapsto a t$ of $\mathbf{R}$ into F is continuous.