Exercise 7, § 2

Given $n$ points $x_i (1 \leq i \leq n)$ in a Hausdorff prehilbertian space E, we mean by the Gram’s determinant of these $n$ points the determinant

a) Show that $G(x_1, ..., x_n) \geq 0$ and that for $x_1, ..., x_n$ to form an independent system, it is necessary and sufficient that $G(x_1, ..., x_n) \neq 0$ (assuming that $\dim(E) \geq n$, consider an orthonormal basis of an $n$-dimensional subspace containing $x_1, ..., x_n$). b) Show that if $x_1, ..., x_n$ is an independent system in E, the distance from a point $x \in E$ to the vector subspace V generated by $x_1, ..., x_n$ is equal to $(G(x, x_1, ..., x_n)/G(x_1, ..., x_n))^{1/2}$ (find the expression for the orthogonal projection of $x$ on V). c) Let $(x_n)$ be an infinite sequence of points in E. For the family $(x_n)$ to be topologically independent, it is necessary and sufficient that, for every integer $p > 0$,

(use $b$).