For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a singular vector subspace of $E$. We assume that $(e_1, \ldots, e_s)$ is a basis of $F \cap F^\perp$. We denote by $G$ a supplementary subspace of $F \cap F^\perp$ in $F$.
a) Show that $G$ is non-singular.
b) Demonstrate by induction on the dimension of $F \cap F^\perp$ (starting with $\operatorname{dim}(F \cap F^\perp) = 1$, then $\operatorname{dim}(F \cap F^\perp) > 1$) that there exist $s$ planes $P_1, \ldots, P_s$ of $E$ such that the following three properties are verified:

1. For all $i \in \{1,\ldots,s\}$, $(P_i, q_{/P_i})$ is an artinian plane containing $e_i$.
2. For all $(i,j) \in \{1,\ldots,s\}^2$ with $i \neq j$, $P_i$ is orthogonal to $P_j$.
3. For all $i \in \{1,\ldots,s\}$, $P_i$ is orthogonal to $G$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that $\bar{F} = G \oplus P_1 \oplus \ldots \oplus P_s$ is non-singular. We will say that $\bar{F}$ is a non-singular completion of $F$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that if $q_{/F} = 0$, then $\operatorname{dim}(F) \leq \frac{n}{2}$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $n = 2p$. Show that $(E,q)$ is an Artin space if and only if there exists a vector subspace $F$ of $E$ of dimension $p$ such that $q_{/F} = 0$.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself, that is, the set of automorphisms $f$ of $E$ satisfying: for all $x \in E$, $q(f(x)) = q(x)$.
Let $f$ be an endomorphism of $E$.
a) Show that $f \in O(E,q)$ if and only if, for all $(x,y) \in E^2$: $\varphi(f(x),f(y)) = \varphi(x,y)$. Show that if $F$ is a vector subspace of $E$ and if $f \in O(E,q)$, then $f(F^\perp) = (f(F))^\perp$.
b) Let $e$ be a basis of $E$. Compute the matrix of the bilinear form: $$(x,y) \mapsto \varphi(f(x),f(y)) \text{ in terms of } \operatorname{mat}(f,e) \text{ and } \operatorname{mat}(\varphi,e).$$
c) Let us set $M = \operatorname{mat}(f,e)$ and $\Omega = \operatorname{mat}(\varphi,e)$. Show that $f \in O(E,q)$ if and only if $\Omega = {}^t M \Omega M$.
d) Show that if $f \in O(E,q)$, then $\operatorname{det}(f) \in \{1,-1\}$. We will denote: $$O^+(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = 1\} \text{ and } O^-(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = -1\}.$$

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ and $G$ be two vector subspaces of $E$ such that $E = F \oplus G$. We denote by $s$ the symmetry with respect to $F$ parallel to $G$.
a) Show that $s \in O(E,q)$ if and only if $F$ and $G$ are orthogonal (for $\varphi$).
b) Deduce that the symmetries in $O(E,q)$ are the symmetries with respect to $F$ parallel to $F^\perp$, where $F$ is a non-singular subspace of $E$.
c) When $H$ is a non-singular hyperplane, we will call reflection along $H$ the symmetry with respect to $H$ parallel to $H^\perp$. Show that every reflection of $E$ is an element of $O^-(E,q)$.
d) Let $(x,y) \in E^2$ such that $q(x) = q(y)$ and $q(x-y) \neq 0$. We denote by $s$ the reflection along $H = \{x-y\}^\perp$. Show that $s(x) = y$.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Suppose that $E$ is an artinian space of dimension $2p$ and that $F$ is a subspace of $E$ of dimension $p$ such that $q_{/F} = 0$.
If $f \in O(E,q)$ with $f(F) = F$, show that $f \in O^+(E,q)$.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ be a subspace of $E$ such that $\bar{F} = E$ (where $\bar{F}$ is a non-singular completion of $F$). Show that if $f \in O(E,q)$ with $f_{/F} = \operatorname{Id}_F$ (where $\operatorname{Id}_F$ is the identity application from $F$ to $F$), then $f \in O^+(E,q)$.

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Show that the map $(P, Q) \mapsto \langle P, Q \rangle$ is an inner product on $\mathbb{R}[X]$.

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Determine a family of polynomials $\left(K_n\right)_{n \in \mathbb{N}}$ satisfying the following two conditions:
i. for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ and its leading coefficient is strictly positive;
ii. for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for the inner product $\langle \cdot, \cdot \rangle$.
Justify the uniqueness of such a family.

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$ The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the unique family of polynomials such that for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
Calculate $K_0$, $K_1$ and $K_2$.

The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the unique family of polynomials such that for all $p \in \mathbb{N}$, the degree of $K_p$ equals $p$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_p\right)_{0 \leqslant p \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
For all $p \in \llbracket 0; n-1 \rrbracket$, calculate $K_p(1)$.

Let $\lambda$ be a real number in the interval $]0,1[$, and let $a$ and $b$ be two non-negative real numbers. Show that $$\lambda a + (1-\lambda) b \geq a^{\lambda} b^{1-\lambda}$$ (one may introduce a certain auxiliary function and justify its concavity). Moreover, show that for all real $u > 1$, $$(\lambda a + (1-\lambda) b)^{u} \leq \lambda a^{u} + (1-\lambda) b^{u}$$

Prove that the real symmetric matrix $A$ is invertible. (One may consider the kernel of the map $x \mapsto A x$).

Let $a$ and $b$ be two non-negative real numbers and $\lambda$ a real number in $]0,1[$. Show that $$(a+b)^{\lambda} \leq a^{\lambda} + b^{\lambda}$$

Prove that $\forall x , y \in \mathbb { R } ^ { n } , \left\langle A ^ { - 1 } x ; y \right\rangle = \left\langle x ; A ^ { - 1 } y \right\rangle$. Deduce that the matrix $A ^ { - 1 }$ is symmetric.

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
We denote $F = \int_{-\infty}^{+\infty} f(x)\,dx$ and $G = \int_{-\infty}^{+\infty} g(x)\,dx$. Show that for all $t$ in the interval $]0,1[$ there exists a unique real number denoted $u(t)$ and a unique real number denoted $v(t)$ such that $$\frac{1}{F} \int_{-\infty}^{u(t)} f(x)\,dx = t, \quad \frac{1}{G} \int_{-\infty}^{v(t)} g(x)\,dx = t$$ (One may study the variations of the function: $u \mapsto \frac{1}{F} \int_{-\infty}^{u} f(x)\,dx$).

For $x , y \in \mathbb { R } ^ { n }$, we set: $( x ; y ) _ { A } = \langle A x ; y \rangle$. We denote by $E$ the endomorphism of the vector space $\mathbb { R } ^ { n }$ defined by $\forall x \in \mathbb { R } ^ { n } , E ( x ) = A ^ { - 1 } K x$.
Prove that $( ; ) _ { A }$ defines an inner product on $\mathbb { R } ^ { n }$. Then show that $$\forall x , y \in \mathbb { R } ^ { n } , ( E ( x ) ; y ) _ { A } = ( x ; E ( y ) ) _ { A } .$$

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the applications $u$ and $v$ are of class $C^{1}$ on the interval $]0,1[$ and, for each $t \in ]0,1[$, calculate the derivatives $u'(t)$ and $v'(t)$.

Show that there exists a basis $\left( e _ { i } \right) _ { 1 \leq i \leq n }$ of $\mathbb { R } ^ { n }$ and $n$ strictly positive real numbers $\lambda _ { i } \in \mathbb { R } ^ { + * } ( 1 \leq i \leq n )$ such that $$\forall i \in \{ 1 , \ldots , n \} , A ^ { - 1 } K e _ { i } = \lambda _ { i } e _ { i }$$

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the image set of the application $w$ defined on $]0,1[$ by $$\forall t \in ]0,1[, \quad w(t) = \lambda u(t) + (1-\lambda) v(t),$$ is equal to $\mathbb{R}$. Then prove that $w$ defines a change of variable from $]0,1[$ to $\mathbb{R}$. Using this and $\int_{-\infty}^{+\infty} h(w)\,dw$, show that $f$, $g$ and $h$ satisfy the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}.$$

We set $\Psi(u) = \exp(-u^{2})$ for all real $u$. Prove that for all $x, y \in \mathbb{R}$, $$\Psi(\lambda x + (1-\lambda) y) \geq \Psi(x)^{\lambda} \Psi(y)^{1-\lambda}$$

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set: $$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$
Let $x, y \in \mathbb{R}$. We set $z = \lambda x + (1-\lambda) y$. Prove that if $|y| \leq M$ then $\Psi(x) \leq \Psi_{M}(z)$. Similarly, prove that if $|x| \leq M$ then $\Psi(y) \leq \Psi_{M}(z)$.

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set: $$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$
Let $\epsilon \in ]0,1[$, $f_{\epsilon} = f + \epsilon\Psi$ and $g_{\epsilon} = g + \epsilon\Psi$. Show that $$\forall x, y \in \mathbb{R}, \quad f_{\epsilon}(x)^{\lambda} g_{\epsilon}(y)^{1-\lambda} \leq h(z) + \epsilon^{\Lambda}\left(\|f\|_{\infty}^{\lambda} + \|g\|_{\infty}^{1-\lambda}\right)\left(\Psi_{M}(z)\right)^{\Lambda} + \epsilon\Psi(z)$$ where $z = \lambda x + (1-\lambda) y$. One should begin by applying the inequality from question 2, then the two preceding questions. (We recall that $f(x) = 0$ if $|x| > M$ and that $g(y) = 0$ if $|y| > M$).

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. Deduce that if $f$ and $g$ are zero outside a bounded interval then the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}$$ is satisfied.