Proposition 7

The space $P_{n,p}$ is compact.

It is enough to show that there is a compact subspace of $L_{n+1,p+1}$ which meets every equivalence class mod $\Delta_{n,p}$ in at least one point; for $P_{n,p}$ is then the image of this subspace under the canonical mapping of $L_{n+1,p+1}$ onto $P_{n,p}$, and is therefore compact (Chapter I, § 9, no. 4, Theorem 2).

Let $V_{n+1,p+1}$ be the subspace of $L_{n+1,p+1}$ whose elements are the systems $(x_k)$ of $p+1$ vectors forming an orthonormal Euclidean basis of the vector subspace they generate that is to say such that $(x_h/x_k) = 0$ whenever $h \neq k$, $(x_h/x_k) = 1$ for $1 \leq h \leq p+1$. Every vector subspace of $p+1$ dimensions of $\mathbf{R}^{n+1}$ has such a basis and therefore every class mod. $\Delta_{n,p}$ meets $V_{n+1,p+1}$. On the other hand, the matrices $X = (x_{ij})$ of $V_{n+1,p+1}$ are defined by the relations

therefore they form a closed set in $M_{p+1,n+1}$; and since these relations imply that $|x_{ij}| \leq 1$ for each pair of indices $(i, j)$, this set is bounded and hence compact.