For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \det(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all family $e \in E^p$. In the sequel for all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$$

For every $x \in E$, we denote by $\omega ( x , \cdot )$ the linear map from $E$ to $\mathbb { R }$, $y \mapsto \omega ( x , y )$ and we consider
$$d _ { \omega } : \left\lvert \, \begin{array} { c c c } E & \rightarrow & \mathcal { L } ( E , \mathbb { R } ) \\ x & \mapsto & \omega ( x , \cdot ) \end{array} \right.$$
Show that $d _ { \omega }$ is an isomorphism.

For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.

For every $x \in E$, we denote by $\omega ( x , \cdot )$ the linear map from $E$ to $\mathbb { R }$, $y \mapsto \omega ( x , y )$ and we consider
$$d _ { \omega } : \left\lvert \, \begin{array} { c c c } E & \rightarrow & \mathcal { L } ( E , \mathbb { R } ) \\ x & \mapsto & \omega ( x , \cdot ) \end{array} \right.$$
Show that $d _ { \omega }$ is an isomorphism.

For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.

Show that, if $M \in \mathcal { M } _ { n } ( \mathbb { R } )$ is nilpotent, then $M ^ { 2 }$ is nilpotent.

Prove that the map $$\begin{array}{ccl} \Psi : \mathbb{R}_{K-1}[X] & \rightarrow & \mathbb{R}^{K} \\ P & \mapsto & \left(P\left(x_{1}\right), \ldots, P\left(x_{K}\right)\right) \end{array}$$ is an isomorphism of vector spaces.

Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0, +\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0, +\infty[$.

Prove that the map $$\begin{array}{ccl} \Psi : \mathbb{R}_{K-1}[X] & \rightarrow & \mathbb{R}^K \\ P & \mapsto & \left(P\left(x_1\right), \ldots, P\left(x_K\right)\right) \end{array}$$ is an isomorphism of vector spaces.

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0,+\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0,+\infty[$.

Show that $\langle \cdot , \cdot \rangle$ is an inner product on $\mathbb { R } _ { n - 1 } [ X ]$, where $$\langle P , Q \rangle = \sum _ { k = 1 } ^ { n } P \left( a _ { k } \right) Q \left( a _ { k } \right).$$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are odd.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are periodic with period 1.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $$\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}.$$
Show that $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are odd.

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are periodic with period 1.

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.
Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$, $$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$

We fix a Markov kernel $K$. Show that for all $n \in \mathbf{N}$, $K^n$ is a Markov kernel.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set $$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$
Show that $\|\cdot\|_{1} : \varphi \mapsto \|\varphi\|_{1}$ is a norm on $\mathcal{C}_{c}(\mathbb{R})$. One may admit without proof that $\|\cdot\|_{\infty}$ is also a norm.

Let $K$ be a compact set of $\mathbb{R}$. Let $k > 0$ and $B$ the set of functions from $K$ to $\mathbb{R}^d$ that are $k$-Lipschitz. Show that $B$ is equicontinuous.

Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$. Show that the map $\mathscr { D } _ { \rho } ( \mathbb { R } ) \rightarrow \mathscr { D } _ { r } ( \mathbb { R } )$ which associates to a function $f$ its restriction to $U _ { r }$ is injective.

Let $n , m \in \mathbb { N }$ and let $r , s \in \mathbb { R } _ { + } ^ { * } , r < \rho$.
8a. Show that $\| \cdot \| _ { r , s }$ is a norm on $\mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
8b. Show that if $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { m } [ X ] \right)$, then $P Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n + m } [ X ] \right)$ and $$\| P Q \| _ { r , s } \leqslant \| P \| _ { r , s } \cdot \| Q \| _ { r , s }$$