Exercise 5, § 6

Let E be the vector space of quadratic forms on $\mathbf{R}^n$, which is identified with the vector subspace of symmetric square matrices in the space $\mathbf{M}_n(\mathbf{R})$ of square matrices of order $n$ on $\mathbf{R}$. We endow $\mathbf{M}_n(\mathbf{R})$ with the scalar product $\operatorname{Tr}(X \cdot Y)$, which enables us to identify it with its dual and similarly for E. a) Let $P \subset E$ be the set of quadratic forms for which the matrix has all elements $\geq 0$, and let $S \subset E$ be the set of positive quadratic forms in $\mathbf{R}^n$. Show that we have $P = P^\circ$ and $S = S^\circ$. b) Let B be the set of quadratic forms on $\mathbf{R}^n$ that can be written in the form $\sum_{j=1}^m {x_j'}^2$ for same $m$, where $x_j'$ is a linear form that takes values $\geq 0$ for all $x = (x_i)_{1 \leq i \leq n}$ of coordinates $x_i$ all $\geq 0$; let C be the set of quadratic forms that are $\geq 0$ for all the vectors $x = (x_i)$ with coordinates $x_i$ all $\geq 0$. Show that $B = C^\circ$ and $C = B^\circ$ (prove that B is closed, showing that every element of B can be written in the form $\sum_{j=1}^m {x_j'}^2$, with $x_j'$ positive for all $x$ with coordinates $\geq 0$, and $m \leq 2^n$).