Show that the family $\left( L _ { 1 } , \ldots , L _ { n } \right)$ is an orthonormal basis of $\mathbb { R } _ { n - 1 } [ X ]$ equipped with the inner product $\langle \cdot , \cdot \rangle$.

Show that, for all $x \in \mathbb { R } _ { + } ^ { * }$, the function $\left\lvert\, \begin{array} { r l l } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ t & \mapsto & \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } \end{array} \quad \frac { \mathrm { e } ^ { - t } } { t } \right.$ is integrable on $\left. ] 0 , x \right]$.

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 $D = f - g$.
Deduce that the function $\widetilde{D}$ is zero on $\mathbb{R}$, then that: $$\forall x \in \mathbb{R} \backslash \mathbb{Z}, \quad \pi x \operatorname{cotan}(\pi x) = 1 + 2\sum_{n=1}^{+\infty} \frac{x^2}{x^2 - n^2}.$$

Show that for all $u,v\in\mathcal{H}$, we have $$\operatorname{ch}(d(u,v)) = -B(u,v).$$

Show that the application $\phi : (P, Q) \mapsto \phi(P, Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ defines an inner product on $\mathbb{R}_{n-1}[X]$.

In what follows, for all $z \in D$ we denote $$P(z) := \exp\left[\sum_{n=1}^{+\infty} L(z^n)\right].$$ Let $z \in D$. Verify that $P(z) \neq 0$, that $$P(t) = \lim_{N \rightarrow +\infty} \prod_{n=1}^{N} \frac{1}{1-t^n}$$ and that for all real $t > 0$, $$\ln P(e^{-t}) = -\sum_{n=1}^{+\infty} \ln(1-e^{-nt}).$$

$\mathbf{5}$ ▷ Show that $\operatorname{det}\left(e^{A}\right) = e^{\operatorname{tr}(A)}$.

Let $N$ be a matrix in $M_{n}(\mathbf{R})$ similar to an almost diagonal matrix. Prove that $N$ is semi-simple.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathscr{A}_p(\mathbb{R}^p, \mathbb{R})$.
(b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathscr{A}_p(F, \mathbb{R})$.

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that $D - C$ is a convex closed subset of $\mathbb{R}^d$ not containing $0$.

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Conclude the case $K = 2$ by showing the interpolation inequality $$\forall f \in \mathcal{C}^{2}([0,1]), \quad \max\left(\|f\|_{\infty}, \left\|f^{\prime}\right\|_{\infty}\right) \leqslant \left\|f^{\prime\prime}\right\|_{\infty} + C\left(\left|f\left(x_{1}\right)\right| + \left|f\left(x_{2}\right)\right|\right)$$ with $C = 1 + \frac{1}{x_{2} - x_{1}}$.

We consider the vector $$w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Show that if $B(v,w_3)<0$, then $d(v_0, s_{w_3}(v)) < d(v_0,v)$.

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 every integer $k\geq 1$, recall that $$P_k = \{v\in\mathcal{H}\cap V_\mathbb{Q} \text{ such that } kv\in V_\mathbb{Z}\}.$$ Show that the set $P_k$ is invariant under $\Gamma$.

We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ Using the result of Question 7, deduce that $f \in E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

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 $\mathscr{M}(E)$ the set of probability measures on $E$. If $\mu \in \mathscr{M}(E)$, we denote by $\mu(x)$ the value $\mu(\{x\})$.
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 $\mathscr{M}(E)$ is a subset of $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

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 $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. We denote $\mathscr{P}(E)$ 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 $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that $\mathscr{M}(E)$ is a subset of $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

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 $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
We denote $$\mathcal{A} = \left\{H \text{ vector subspace of } E \text{ such that } u(H) \subset H \text{ and } F \cap H = \{0_{E}\}\right\}$$ and $$\mathcal{L} = \left\{p \in \mathbf{N}^{*} \mid \exists H \in \mathcal{A} : p = \operatorname{dim}(H)\right\}$$
Prove that $F$ has a complement $G$ in $E$, stable under $u$.

Let $F$ be a vector subspace of a symplectic space $(E,\omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega(x,y) = 0 \}$. Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $$\left\|f^{(k)} - P^{(k)}\right\|_{\infty} \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_{\infty}$$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Show that, if $f$ and $g$ are two functions in $E$, then the integral $\int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is absolutely convergent, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

Let $(c_{j})_{j \in \mathbf{N}}$ be a sequence of complex numbers such that the series $\sum c_{j}$ converges absolutely. We set $$\forall t \in \mathbf{R}, \quad u(t) = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$
Justify the existence and continuity of the function $u$. For $k \in \mathbf{N}$, show that $$\frac{1}{2\pi} \int_{-\pi}^{\pi} u(t) e^{i(k+1)t} \mathrm{d}t = c_{k}$$

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 $\mathscr{M}(E)$ the set of probability measures on $E$. 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 $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and let $\mu$ be an element of $\mathscr{M}(E)$. Show that if the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in the normed vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$, then $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x).$$