Proposition 10

— In the space $\mathscr{L}_F^p$, the linear subspace formed by the (finite) linear combinations $\sum_k a_k f_k$, where $a_k \in F$ and the $f_k$ are continuous numerical functions with compact support, is dense (for the topology of convergence in mean of order $p$).

The set $\mathscr{K}_F$ of continuous mappings of X into F with compact support is by definition dense in $\mathscr{L}_F^p$. On the other hand, every function $g \in \mathscr{K}_F$ may be uniformly approximated by functions of the form $\sum_k a_k f_k$, where the $f_k$ are continuous functions with support contained in a fixed compact neighborhood of the support of $g$ (Ch. III, §1, No. 2, Lemma 2); it follows (No. 3, Prop. 4) that $g$ is in the closure in $\mathscr{L}_F^p$ of the set of $\sum_k a_k f_k$, whence the proposition.