Let $m \in \mathbb{N}^{\star}$ and $(u_1, \ldots, u_m)$ be a family of vectors of $\mathbb{R}^n$. We denote $$C = \left\{ \sum_{i=1}^{m} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in \llbracket 1, m \rrbracket \right\}.$$ The purpose of this question is to show that $C$ is a closed convex set of $\mathbb{R}^n$. a. Show that $C$ is convex. b. Show that if $(u_1, \ldots, u_m)$ is a free family, then $C$ is closed. c. For all $I \subset \llbracket 1, m \rrbracket$, we set $C_I = \left\{ \sum_{i \in I} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in I \right\}$. Show that $$C = \bigcup_{I} C_I,$$ where the union is taken over the sets $I \subset \llbracket 1, m \rrbracket$ such that $(u_i)_{i \in I}$ is a free family. Deduce that $C$ is closed.
Let $m \in \mathbb{N}^{\star}$ and $(u_1, \ldots, u_m)$ be a family of vectors of $\mathbb{R}^n$. We denote
$$C = \left\{ \sum_{i=1}^{m} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in \llbracket 1, m \rrbracket \right\}.$$
The purpose of this question is to show that $C$ is a closed convex set of $\mathbb{R}^n$.\\
a. Show that $C$ is convex.\\
b. Show that if $(u_1, \ldots, u_m)$ is a free family, then $C$ is closed.\\
c. For all $I \subset \llbracket 1, m \rrbracket$, we set $C_I = \left\{ \sum_{i \in I} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in I \right\}$. Show that
$$C = \bigcup_{I} C_I,$$
where the union is taken over the sets $I \subset \llbracket 1, m \rrbracket$ such that $(u_i)_{i \in I}$ is a free family. Deduce that $C$ is closed.