Proposition 9

— Let F be an n-dimensional vector space over $\mathbf{R}$, and let $(e_k)_{1 \leq k \leq n}$ be a basis of F. For a function $f = \sum_{k=1}^n e_k f_k$ to belong to $\mathscr{L}_F^p$, it is necessary and sufficient that each of the numerical functions $f_k$ belong to $\mathscr{L}^p$.

This follows at once from Cors. 1 and 2 of Th. 4.