Let $A \in \mathrm { M } _ { 2 } ( \mathrm { C } )$ be a matrix and let $\eta$ be a strictly positive real number.
(a) For $x \in \mathrm { C } ^ { 2 }$, show that the series $$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ is convergent. We denote $$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ the sum of this series.
(b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality $$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x )$$ (c) Show that there exists a real $C > 0$ such that for every $x \in \mathbb { C } ^ { 2 }$ we have $$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$

(a) If $B \in \mathbf { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for every $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathbf { G L } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$.
(b) Show that there exists a matrix $C \in \mathbf { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.

Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathbf { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies $$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 } .$$ Show that there exists $\varepsilon > 0$ such that for every $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$ ) converges to $x ^ { * }$ when $n \rightarrow + \infty$.

Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\eta$ be a strictly positive real number.
(a) For $x \in \mathbb { C } ^ { 2 }$, show that the series $$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ is convergent.
We denote $$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ the sum of this series.
(b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality $$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x ) .$$
(c) Show that there exists a real $C > 0$ such that for all $x \in \mathbb { C } ^ { 2 }$ we have $$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$

(a) If $B \in \mathrm { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for all $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathrm { GL } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$.
(b) Show that there exists a matrix $C \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.

Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathrm { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies $$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 }$$ Show that there exists $\varepsilon > 0$ such that for all $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$) converges to $x ^ { * }$ when $n \rightarrow + \infty$.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$. Let $y \in \mathbb{R}^d$, show that $$y = \operatorname{proj}_C(x) \Longleftrightarrow y \in C \text{ and } (x - y) \cdot (z - y) \leqslant 0, \forall z \in C.$$

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$. Let $y \in \mathbb{R}^d$, show that $$y = \operatorname{proj}_C(x) \Longleftrightarrow y \in C \text{ and } (x - y) \cdot (z - y) \leqslant 0, \forall z \in C.$$

Let $C \in ]0,1[$. Using a simple example of a function $f$, show that the interpolation inequality $$\forall f \in \mathcal{C}^{1}([0,1]), \quad \|f\|_{\infty} \leqslant \left\|f^{\prime}\right\|_{\infty} + C\left|f\left(x_{1}\right)\right|$$ is false.

Let $C \in ]0,1[$. Using a simple example of a function $f$, show that the interpolation inequality $$\forall f \in \mathcal{C}^1([0,1]), \quad \|f\|_\infty \leqslant \left\|f^\prime\right\|_\infty + C\left|f\left(x_1\right)\right|$$ is false.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Determine explicitly $\operatorname{proj}_C$ in the following cases: $$\text{i) } C = \mathbb{R}_+^d, \quad \text{ii) } C = \left\{y \in \mathbb{R}^d : \|y\| \leqslant 1\right\}$$ $$\text{iii) } C = \left\{y \in \mathbb{R}^d : \sum_{i=1}^d y_i \leqslant 1\right\}, \quad \text{iv) } C = [-1,1]^d$$

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$. The $\omega$-orthogonal of $F$ is defined as $$F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}.$$ Justify that $F ^ { \omega }$ is a vector subspace of $E$.

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$. The $\omega$-orthogonal of $F$ is defined as $$F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$$ Justify that $F ^ { \omega }$ is a vector subspace of $E$.

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Deduce that, for any function $f \in \mathcal{C}^{2}([0,1])$, we have $$\left\|f^{\prime}\right\|_{\infty} \leqslant \left\|f^{\prime\prime}\right\|_{\infty} + \frac{\left|f\left(x_{1}\right)\right| + \left|f\left(x_{2}\right)\right|}{x_{2} - x_{1}}.$$

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Deduce that, for any function $f \in \mathcal{C}^2([0,1])$, we have $$\left\|f^\prime\right\|_\infty \leqslant \left\|f^{\prime\prime}\right\|_\infty + \frac{\left|f\left(x_1\right)\right| + \left|f\left(x_2\right)\right|}{x_2 - x_1}.$$

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 $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded.
Assume that $n \geq 2$. Indicate, with justification, a matrix $M$ in $\mathcal{M}_{n}(\mathbf{C})$, upper triangular, such that $\sigma(M) \subset \mathbb{D}$, but not belonging to $\mathcal{B}_{n}$.

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: $$\forall x \in \mathbb{R} \backslash \mathbb{Z}, \quad \pi x \operatorname{cotan}(\pi x) = 1 + 2\sum_{n=1}^{+\infty} \frac{x^2}{x^2 - n^2}.$$

Let $D = f - g$ where $f(x) = \pi \operatorname{cotan}(\pi x)$ and $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and let $\widetilde{D}$ be its continuous extension to $\mathbb{R}$. Deduce that the function $\widetilde{D}$ is zero on $\mathbb{R}$, then that: $$\forall x \in \mathbb{R} \backslash \mathbb{Z}, \quad \pi x \operatorname{cotan}(\pi x) = 1 + 2\sum_{n=1}^{+\infty} \frac{x^2}{x^2 - n^2}$$

We denote by arcch $:[1,+\infty)\rightarrow\mathbb{R}_+$ the inverse of the hyperbolic cosine. Let $v\in\mathcal{H}$. Show that the set $T_v\mathcal{H}$ of vectors tangent to $\mathcal{H}$ at point $v$ is a vector subspace of $V$ and determine this subspace. Deduce that the restriction of $B$ to $T_v\mathcal{H}$ is an inner product.

Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by $$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$ Show that if $h:[c,d]\rightarrow[a,b]$ is a diffeomorphism, then $\ell(\gamma) = \ell(\gamma\circ h)$.

Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by $$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$ Let $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$. Show that $$f'(t) \leq \sqrt{f(t)^2-1}\, n(t).$$

Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$, with $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$ satisfying $f'(t) \leq \sqrt{f(t)^2-1}\,n(t)$.
Deduce that $$-B(\gamma(a),\gamma(b)) \leq \operatorname{ch}(\ell(\gamma)).$$

Let $u$ and $v$ be two points of $\mathcal{H}$. The hyperbolic distance between $u$ and $v$ is defined by $$d(u,v) = \inf_\gamma \ell(\gamma)$$ where the infimum is taken over the set of continuous and piecewise $\mathcal{C}^1$ paths $\gamma:[a,b]\rightarrow\mathcal{H}$ such that $\gamma(a)=u$ and $\gamma(b)=v$.
Show that $d$ is a distance on $\mathcal{H}$, that is,

• $d(u,v) = d(v,u)$,
• $d(u,w) \leq d(u,v)+d(v,w)$, and
• $d(u,v)=0 \Leftrightarrow u=v$

for all $u,v,w\in\mathcal{H}$.