We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that $T$ is injective.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that if $\lambda \in \sigma_p(T)$ and $f \in \operatorname{Ker}(T - \lambda Id)$, then $f \in C^2([0,1], \mathbb{R})$ and satisfies the equation $$\lambda f'' + f = 0$$ with the conditions $f(0) = f(1) = 0$.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Deduce $\sigma_p(T)$. Calculate the eigenspaces $E_{\lambda} = \operatorname{Ker}(T - \lambda Id)$ associated with each element $\lambda \in \sigma_p(T)$.

Let $q \in \mathcal { Q } ( V )$ be a non-degenerate quadratic form on $V$.
(a) Let $V ^ { \prime }$ be a $\mathbb { K }$-vector space of finite dimension and $q ^ { \prime }$ a quadratic form on $V ^ { \prime }$. Prove that if $q$ and $q ^ { \prime }$ are isometric, then $q ^ { \prime }$ is in $\mathcal { Q } \left( V ^ { \prime } \right)$, that is, non-degenerate.
(b) For $x \neq 0$, we denote $\{ x \} ^ { \perp } : = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$. Show that $\{ x \} ^ { \perp }$ is a vector subspace of $V$ of dimension $n - 1$.
(c) Under what condition on $x$ is the subspace $\{ x \} ^ { \perp }$ a complement of the line $\mathbb { K } x$ in $V$ ?

Application. For all $n \in \mathbb { N } ^ { * }$, we denote $\Gamma ( n ) = \int _ { 0 } ^ { + \infty } x ^ { n - 1 } e ^ { - x } d x$.
(a) Calculate $\Gamma ( n )$ for all $n \in \mathbb { N } ^ { * }$. One will use induction.
(b) Deduce the following asymptotic equivalent $$n ! \underset { n \rightarrow + \infty } { \sim } \sqrt { 2 \pi } n ^ { n + 1 / 2 } e ^ { - n }$$ Hint. First rewrite $\Gamma ( n + 1 )$ in the form $$\Gamma ( n + 1 ) = n ^ { n + 1 } \int _ { 0 } ^ { + \infty } e ^ { - n ( x - \ln x ) } d x$$

We recall that a pre-Hilbert space $H$ is a normed vector space whose norm is derived from an inner product denoted $\langle .,. \rangle$. We call Hilbert basis of $H$ any family $B = (b_i)_{i \in \mathbb{N}}$ such that: (i) the family is orthonormal: for all $i$ and $j$ in $\mathbb{N}$, $\langle b_i, b_j \rangle = 1$ if $i = j$ and $0$ otherwise. (ii) every element $x$ of $H$ can be written: $x = \sum_{i=0}^{+\infty} \langle x, b_i \rangle b_i$, that is $$\lim_{N \rightarrow +\infty} \left\| x - \sum_{i=0}^{N} \langle x, b_i \rangle b_i \right\| = 0$$
Show that if $B = (b_i)_{i \in \mathbb{N}}$ is a Hilbert basis of $H$, then $$\forall x \in H, \quad \|x\|^2 = \sum_{i=0}^{+\infty} |\langle x, b_i \rangle|^2$$

Show that $H = l^2(\mathbb{N})$ equipped with the norm $\|.\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$ is a pre-Hilbert space for the inner product: $$\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$$ (justify that this is indeed an inner product) then determine a Hilbert basis of $H$.

In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $T$ be an operator on $H$. We admit the existence of an operator $\tilde{T} \in \mathcal{L}(H)$ such that $$\forall (x,y) \in H^2, \quad \langle T(x), y \rangle = \langle x, \tilde{T}(y) \rangle$$ Let $B = (b_i)_{i \in \mathbb{N}}$ and $C = (c_i)_{i \in \mathbb{N}}$ be two Hilbert bases of $H$ such that $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty$$ Show that $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 = \sum_{i=0}^{+\infty} \|\tilde{T}(c_i)\|^2$$

In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $B = (b_i)_{i \in \mathbb{N}}$ be a Hilbert basis of $H$ and $T \in \mathcal{L}(H)$. Show that the quantity (possibly infinite) $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2$$ does not depend on the basis $B$. We denote $$\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$$ and we set $$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$$

In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, with $S$ the left shift $(Su)_n = u_{n-1}$ if $n \geq 1$, $(Su)_0 = 0$, and $V$ the right shift $(Vu)_n = u_{n+1}$, and $$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty\right\}$$ Show that the operators $S$ and $V$ defined in part 2 are not in $\mathcal{L}^2(H)$. Give an example of a non-zero operator in $\mathcal{L}^2(H)$.

With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
Show that $\mathcal{L}^2(H)$ equipped with $\|.\|_2$ has the structure of a normed vector space.

With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
Let $L$ and $U$ be in $\mathcal{L}^2(H)$ and $B = (b_i)_{i \in \mathbb{N}}$ a Hilbert basis of $H$. Show that the quantity $$\sum_{i=0}^{+\infty} \langle L(b_i), U(b_i) \rangle$$ is finite, independent of the basis $B$ chosen, and defines an inner product on $\mathcal{L}^2(H)$.

With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
We consider $L$ and $U$ two operators in $\mathcal{L}(H)$. Show that if $L \in \mathcal{L}^2(H)$, then so is $UL$.

With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
We consider $L$ and $U$ two operators in $\mathcal{L}(H)$. What happens for $UL$ assuming this time that $U \in \mathcal{L}^2(H)$?

Let $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ where $V ^ { \prime }$ is a $\mathbb { K }$-vector space of finite dimension. Prove that $O ( q )$ is a subgroup of $\mathrm { GL } ( V )$ and that if $q \cong q ^ { \prime }$, then $O ( q )$ and $O \left( q ^ { \prime } \right)$ are two isomorphic groups.

Fourier series. Let $\phi : \mathbb { R } \rightarrow \mathbb { C }$ be a periodic function with period $2 \pi$, of class $\mathcal { C } ^ { 1 }$.
(a) Show that for all $n \in \mathbb { Z } ^ { * } , c _ { n } ( \phi ) = \frac { c _ { n } \left( \phi ^ { \prime } \right) } { \text { in } }$.
(b) Show that the series $\sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$ converges. Hint. Use Parseval's formula for the function $\phi ^ { \prime }$.
(c) Show that $\| \phi \| _ { \infty } \leq \sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$.

We say that $q \in \mathcal{Q}(V)$ is isotropic if there exists $x \in V - \{ 0 \}$ such that $q ( x ) = 0$. Otherwise, we say that $q$ is anisotropic.
(a) Prove that there exists $x \in V$ such that $q ( x ) \neq 0$.
(b) We denote by $h$ the quadratic form on $\mathbb { K } ^ { 2 }$ defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (we do not ask you to verify that $h$ is a quadratic form). Show that if $V$ is of dimension two and $q$ is isotropic then $q$ is isometric to $h$.
(c) Prove that if $q \in \mathcal { Q } ( V )$ is isotropic, then $q : V \rightarrow \mathbb { K }$ is surjective.

Let $\psi : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function, periodic with period $2 \pi$. Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. For every parameter $\varepsilon > 0$, we set $$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$ First case. In this question, we further assume that $\psi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R }$ and that $f$ has compact support in $] a , b [$.
(a) Show that for all $\varepsilon > 0$, $$\left| J _ { \varepsilon } - c _ { 0 } ( \psi ) \left( \int _ { a } ^ { b } f ( x ) d x \right) \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \sum _ { n \in \mathbb { Z } ^ { * } } \frac { \left| c _ { n } ( \psi ) \right| } { | n | }$$ Hint. One can reduce to the case where $\int _ { 0 } ^ { 2 \pi } \psi ( y ) d y = 0$.
(b) Deduce the limit of $J _ { \varepsilon }$ as $\varepsilon \rightarrow 0$.

A basis $\left( e _ { 1 } , \ldots , e _ { n } \right)$ of $V$ is said to be orthogonal for $q$ if $\widetilde { q } \left( e _ { i } , e _ { j } \right) = 0$ for all $i \neq j$.
(a) Show that there exists an orthogonal basis for $q$.
Hint: one may consider $\{ x \} ^ { \perp } = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$ and use questions 4c and 6a.
(b) Deduce that there exist $a _ { 1 } , \ldots , a _ { n } \in \mathbb { K } - \{ 0 \}$ such that $q \cong \left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$.

Let $\psi : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function, periodic with period $2 \pi$. Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. For every parameter $\varepsilon > 0$, we set $$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$ Second case. We now assume only that $\psi \in \mathcal { C } ^ { 0 } ( \mathbb { R } )$ is periodic with period $2 \pi$, and $f \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. Let $\varepsilon > 0$. We define a subdivision of the interval $[ a , b ]$ as follows. We denote $N _ { \varepsilon }$ the integer part of $\frac { b - a } { 2 \pi \varepsilon }$. We then define $$x _ { k } ^ { \varepsilon } = a + 2 k \pi \varepsilon , \text { for every integer } k \text { such that } 0 \leq k \leq N _ { \varepsilon } .$$ (a) Show that $\lim _ { \varepsilon \rightarrow 0 } x _ { N _ { \varepsilon } } ^ { \varepsilon } = b$.
(b) Deduce that $$\lim _ { \varepsilon \rightarrow 0 } \int _ { x _ { N _ { \varepsilon } } ^ { \varepsilon } } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x = 0$$ (c) Show that for every integer $k$ such that $0 \leq k \leq N _ { \varepsilon } - 1$, for all $x \in \left[ x _ { k } ^ { \varepsilon } , x _ { k + 1 } ^ { \varepsilon } \right]$, $$\left| f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right| \leq 2 \pi \varepsilon \left\| f ^ { \prime } \right\| _ { \infty }$$ (d) Show that $$\sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) f \left( x _ { k } ^ { \varepsilon } \right) d x = \left( \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \varepsilon \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } f \left( x _ { k } ^ { \varepsilon } \right) \right)$$ (e) Show that $$\left| \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) \left( f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right) d x \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \left( \int _ { 0 } ^ { 2 \pi } | \psi ( y ) | d y \right)$$ (f) Deduce that $\lim _ { \varepsilon \rightarrow 0 } J _ { \varepsilon } = \left( \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \int _ { a } ^ { b } f ( x ) d x \right)$.

We assume in this part that $\mathbb { K } = \mathbb { R }$. Let $q \in \mathcal { Q } \left( \mathbb { R } ^ { n } \right) ( n \geq 1 )$. Prove that there exists a pair of integers $( r , s ) ( r + s = n )$ such that $q$ is isometric to $Q _ { r , s }$ defined on the canonical basis of $\mathbb { R } ^ { n }$ by $$Q _ { r , s } \left( x _ { 1 } , \ldots , x _ { n } \right) = \sum _ { i = 1 } ^ { r } x _ { i } ^ { 2 } - \sum _ { j = r + 1 } ^ { n } x _ { j } ^ { 2 }$$

Application. Let $\varepsilon > 0$. Let $\alpha \in \mathbb { R }$. Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function. We consider the following differential equation $$\left\{ \begin{array} { l } u ^ { \prime \prime } ( t ) + u ( t ) = g \left( \frac { t } { \varepsilon } \right) \\ u ( 0 ) = \alpha , u ^ { \prime } ( 0 ) = 0 \end{array} \right.$$ (a) Justify the existence and uniqueness of a solution of (1), defined for $t \in \mathbb { R }$.
(b) Calculate this solution using the method of variation of constants. We denote this solution $u _ { \varepsilon }$.
(c) We assume that $g$ is $2 \pi$-periodic. Show that for all $t \in \mathbb { R } , u _ { \varepsilon } ( t )$ has a limit as $\epsilon \rightarrow 0 ^ { + }$, limit which one will calculate.

We assume $\mathbb{K} = \mathbb{R}$. Let $j : \mathcal { L } \left( \mathbb { R } ^ { n } \right) \longrightarrow \mathcal { M } _ { n } ( \mathbb { R } )$ be the linear isomorphism that associates to every endomorphism its matrix in the canonical basis of $\mathbb { R } ^ { n }$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$.
Let $f : \mathbb { R } ^ { n } \rightarrow \mathbb { R } ^ { n }$ be a linear map and $M = j ( f )$ its matrix in the canonical basis of $\mathbb { R } ^ { n }$. Prove that $M \in O _ { r , s }$ if and only if ${ } ^ { t } M I _ { r , s } M = I _ { r , s }$ where $I _ { r , s }$ is the matrix $$I _ { r , s } = \left[ \begin{array} { c c } I _ { r } & 0 _ { r , s } \\ 0 _ { s , r } & - I _ { s } \end{array} \right]$$ $I _ { p }$ denotes the identity matrix of size $p \times p$ and $0 _ { p , q }$ the zero matrix of size $p \times q$ for all integers $p$ and $q$.
What can be said about the determinant $\operatorname { det } ( M )$ of $M$ if $M \in O _ { r , s }$ ?

In this part, $\varphi : [ a , b ] \rightarrow \mathbb { R }$ and $f : [ a , b ] \rightarrow \mathbb { R }$ are two functions of class $\mathcal { C } ^ { \infty }$. We are interested in integrals of the form $$I ( \lambda ) = \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x$$ where $\lambda$ is a strictly positive real parameter.
Case of a non-stationary phase. We assume in this question that $\varphi ^ { \prime } ( x ) \neq 0$ for all $x \in [ a , b ]$.
(a) We define $L : \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } ) \rightarrow \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ and $M : \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } ) \rightarrow \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ by: for all $g \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$, all $x \in [ a , b ]$, $$L g ( x ) = \frac { 1 } { i \lambda \varphi ^ { \prime } ( x ) } g ^ { \prime } ( x ) , \quad M g ( x ) = - \left( \frac { g } { i \varphi ^ { \prime } } \right) ^ { \prime } ( x )$$ i. Determine the functions $g \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ such that $L g = g$. ii. Let $g , h \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$. We assume that $h$ has compact support in $] a , b [$. Show that $$\int _ { a } ^ { b } h ( x ) L g ( x ) d x = \frac { 1 } { \lambda } \int _ { a } ^ { b } g ( x ) M h ( x ) d x$$ (b) Show that if $f$ has compact support in $] a , b [$, then for all $N \in \mathbb { N } ^ { * }$, there exists a constant $\gamma _ { N }$ independent of $\lambda$ such that $$| I ( \lambda ) | \leq \gamma _ { N } \lambda ^ { - N }$$

We assume $\mathbb{K} = \mathbb{R}$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$. Prove that $O _ { r , s }$ is a closed subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$ (we equip $\mathcal { M } _ { n } ( \mathbb { R } )$, the set of square matrices of size $n$ with coefficients in $\mathbb { R }$, with its topology as a $\mathbb { R }$-vector space of finite dimension).