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 denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $f \in O(E,q)$. We assume that for all $x \in E$ such that $q(x) \neq 0$, we have $f(x) - x \neq 0$ and $q(f(x)-x) = 0$.
We propose to demonstrate that $f \in O^+(E,q)$ and that $E$ is an Artin space.
a) Show that $\operatorname{dim}(E) \geq 3$.
b) We denote by $V = \operatorname{Ker}(f - \operatorname{Id}_E)$. Show that $q_{/V} = 0$.
c) Let $x \in E$ such that $q(x) = 0$. We denote $H = \{x\}^\perp$. Show that $q_{/H}$ is not identically zero. Deduce that there exists $y \in E$ such that $q(x+y) = q(x-y) = q(y) \neq 0$.
d) We denote by $U = \operatorname{Im}(f - \operatorname{Id}_E)$. Show that $q_{/U} = 0$.
e) Show that $U^\perp = V = U$.
f) Deduce that $E$ is an Artin space and that $f \in O^+(E,q)$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
Prove the Cartan-Dieudonné theorem when $n = 1$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude when there exists $x \in E$ such that $f(x) = x$ with $q(x) \neq 0$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude when there exists $x \in E$ such that $q(x) \neq 0$ and $q(f(x)-x) \neq 0$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude in the other cases (i.e., when neither of the conditions in IV.A.2 or IV.A.3 holds).

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 $u > 1$, $$(\lambda a + (1-\lambda) b)^{u} \leq \lambda a^{u} + (1-\lambda) b^{u}$$

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

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.

Let $n \in \mathbb{N}^{*}$. We denote by $\chi_{n} : \mathbb{R} \rightarrow \mathbb{R}$ the continuous function that equals 1 on $[-n, n]$, equals 0 on $]-\infty, -n-1] \cup [n+1, +\infty[$ and is affine on each of the two intervals $[-n-1, -n]$ and $[n, n+1]$.
Show that: $$\forall x, y \in \mathbb{R},\quad \chi_{n}(x)^{\lambda} \chi_{n}(y)^{1-\lambda} \leq \chi_{n+1}(\lambda x + (1-\lambda) y)$$

Show that 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 (if you choose to use the dominated convergence theorem then carefully verify that its conditions of validity are satisfied).

Let $N : \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a norm on the vector space $\mathbb{R}^{n}$. Prove that the application defined by $$\forall x \in \mathbb{R}^{n}, \quad f(x) = \exp\left(-N(x)^{2}\right),$$ is continuous and log-concave on $\mathbb{R}^{n}$. (One may observe that the function $u \mapsto u^{2}$ is convex on $\mathbb{R}_{+}$).

We denote by $C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of continuous functions $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ such that: $$\forall \left( \theta _ { 1 } , \theta _ { 2 } \right) \in \mathbb { R } ^ { 2 } , f \left( \theta _ { 1 } + 2 \pi , \theta _ { 2 } \right) = f \left( \theta _ { 1 } , \theta _ { 2 } \right) = f \left( \theta _ { 1 } , \theta _ { 2 } + 2 \pi \right)$$
Let $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Prove that $$\sup _ { \left( \theta _ { 1 } , \theta _ { 2 } \right) \in \mathbb { R } ^ { 2 } } \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right| = \sup _ { \left( \theta _ { 1 } , \theta _ { 2 } \right) \in [ 0,2 \pi ] ^ { 2 } } \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right|$$ Deduce that $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right|$ is bounded on $\mathbb { R } ^ { 2 }$ and attains its supremum.

Let $\lambda \in ]0,1[$ and $f, g, h$ be functions from $\mathbb{R}^{2}$ to $\mathbb{R}_{+}$ that are continuous with bounded support and such that $$\forall X \in \mathbb{R}^{2}, \forall Y \in \mathbb{R}^{2}, \quad h(\lambda X + (1-\lambda) Y) \geq f(X)^{\lambda} g(Y)^{1-\lambda}$$ Show that $$\iint_{\mathbb{R}^{2}} h(x,y)\,dx\,dy \geq \left(\iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy\right)^{\lambda} \left(\iint_{\mathbb{R}^{2}} g(x,y)\,dx\,dy\right)^{1-\lambda}.$$

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We denote by $C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of functions $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ such that the two partial derivatives $\frac { \partial f } { \partial \theta _ { 1 } } , \frac { \partial f } { \partial \theta _ { 2 } }$ exist at every point of $\mathbb { R } ^ { 2 }$ and both define continuous functions on $\mathbb { R } ^ { 2 }$.
We set $\forall t \in \left[ 0 , + \infty \left[ , \theta ( t ) = \left( t \sqrt { \lambda _ { 1 } } + \varphi _ { 1 } , t \sqrt { \lambda _ { 2 } } + \varphi _ { 2 } \right) \right. \right.$.
The Ergodic Theorem states: Let $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Then, $$\lim _ { T \rightarrow + \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } f \circ \theta ( t ) d t = ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \tag{4}$$
Let $j , l \in \mathbb { Z }$. Prove the Ergodic Theorem in the special case of the function $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto f \left( \theta _ { 1 } , \theta _ { 2 } \right) = e ^ { i \theta _ { 1 } j } e ^ { i \theta _ { 2 } l }$. (In the case where $( j , l ) \neq ( 0,0 )$ one may verify that $j \sqrt { \lambda _ { 1 } } + l \sqrt { \lambda _ { 2 } }$ is non-zero and then one may calculate each side of (4) separately in this special case).