Does the matrix $\Delta _ { p + 1 }$ belong to the set $O ( 1 , p )$ ? to the set $O ^ { + } ( 1 , p )$ ?

Show that, for every matrix $L$ element of $O ( 1 , p )$, its transpose ${ } ^ { t } L$ is also an element of $O ( 1 , p )$.

Show that the sets $O ( 1 , p ) , O ^ { + } ( 1 , p )$ and $O ^ { - } ( 1 , p )$ of $\mathcal { M } _ { p + 1 } ( \mathbb { R } )$ are closed.

Let $A$ and $B$ be two matrices of $\mathcal { M } _ { n } ( \mathbb { R } )$. Show that if, for all $X$ and $Y$ of $\mathbb { R } ^ { n } , { } ^ { t } X A Y = { } ^ { t } X B Y$ then $A = B$.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Express $\varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$ as a function of $q _ { p + 1 } \left( v + v ^ { \prime } \right)$ and $q _ { p + 1 } \left( v - v ^ { \prime } \right)$.

Show that the matrices that are elements of $O ^ { + } ( 1,1 )$ are diagonalizable and find a matrix $P \in O ( 2 )$ such that, for every matrix $L \in O ^ { + } ( 1,1 )$, the matrix ${ } ^ { t } P L P$ is diagonal.

Calculate the exponential of the matrix $M_{p,q,r}$, where $$M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$

Prove that the map $q \mapsto \widetilde { q }$ is a bijection from the set of quadratic forms on $V$ to the set of symmetric bilinear forms on $V$, where $\widetilde { q } : V \times V \rightarrow \mathbb { K }$ is defined by $( x , y ) \mapsto \widetilde { q } ( x , y ) = \frac { 1 } { 2 } ( q ( x + y ) - q ( x ) - q ( y ) )$.

Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Show that $S$ and $V$ belong to $\mathcal{L}(H)$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Show that $S$ and $V$ belong to $\mathcal{L}(F)$.

Let $\mathcal { B } : = \left( e _ { 1 } , \ldots , e _ { n } \right)$ be a basis of $V$. We associate to every symmetric bilinear form $b$ on $V$ a symmetric matrix $\Phi _ { \mathcal { B } } ( b ) : = \left( b \left( e _ { i } , e _ { j } \right) \right) _ { i , j = 1 \ldots n }$ called the matrix of $b$ in the basis $\mathcal { B }$. We recall that $b \mapsto \Phi _ { \mathcal { B } } ( b )$ is an isomorphism between the vector space of symmetric bilinear forms on $V$ and that of square symmetric matrices of size $n$.
(a) Prove that a quadratic form $q$ on $V$ is non-degenerate if and only if the determinant $\operatorname { det } \left( \Phi _ { \mathcal { B } } ( \tilde { q } ) \right)$ is non-zero.
(b) What is the matrix of $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ in the canonical basis of $\mathbb { K } ^ { n }$ ? Deduce that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle \in \mathcal { Q } \left( \mathbb { K } ^ { n } \right)$.

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 $T \in \mathcal{L}(E)$.

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.

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 $V$ ?

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 $U \in \mathcal{L}^2(H)$?

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and $\mathbf{H} = \{I_3 + M \mid M \in \mathbf{L}\}$, with the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that $\mathbf{H}$ equipped with the usual product of matrices is a subgroup of $\mathrm{SL}_3(\mathbf{R})$ and that $$\exp : (\mathbf{L}, *) \rightarrow (\mathbf{H}, \times)$$ is a group isomorphism.

We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$. We further assume that $A$ and $B$ commute with $[A,B]$.
(a) Show that $[A, \exp(B)] = \exp(B)[A,B]$.
(b) Determine a differential equation satisfied by $t \mapsto \exp(tA)\exp(tB)$.
(c) Deduce the formula: $$\exp(A)\exp(B) = \exp\left(A + B + \frac{1}{2}[A,B]\right)$$