Show that $\mathcal{H}(U)$ is a vector subspace of $\mathcal{C}^2(U, \mathbb{R})$.

Show that $\mathcal{H}(U)$ is a vector subspace of $\mathcal{C}^2(U, \mathbb{R})$.

Justify that $\forall k \in \llbracket 1 , n \rrbracket , 0 \leqslant X ^ { k } \leqslant 1 + X ^ { n }$.

Deduce that if $u$, $v$ and $v'$ in $E$ satisfy $v \neq v'$ and $\|u - v\| = \|u - v'\|$ then $\left\|u - \frac{v + v'}{2}\right\| < \|u - v\|$.

Deduce that if $u$, $v$ and $v'$ in $E$ satisfy $v \neq v'$ and $\|u - v\| = \|u - v'\|$ then $\left\|u - \frac{v + v'}{2}\right\| < \|u - v\|$.

Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.

Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
(a) What is the degree of $P_{j}$?
(b) Show that $P_{j}$ is an even or odd polynomial, depending on the value of $j$.
(c) Show that $P_{j}(1) = 1$ and $P_{j}(-1) = (-1)^{j}$.

Let $F$ be a non-empty closed set of $E$ and $u$ in $E$. Show that there exists $v$ in $F$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$

Let $F$ be a non-empty closed set of $E$ and $u$ in $E$. Show that there exists $v$ in $F$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$

We assume in this question that $U$ is path-connected. Determine the set of functions $f$ in $\mathcal{H}(U)$ such that $f^2$ also belongs to $\mathcal{H}(U)$.

We assume in this question that $U$ is path-connected. Determine the set of functions $f$ in $\mathcal{H}(U)$ such that $f^2$ also belongs to $\mathcal{H}(U)$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
By means of integration by parts, show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is orthogonal in $\mathbb{R}_{n}[X]$.

Deduce that if $C$ is a non-empty closed convex set of $E$ and $u$ is a vector of $E$ then there exists a unique $v$ in $C$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$

Deduce that if $C$ is a non-empty closed convex set of $E$ and $u$ is a vector of $E$, then there exists a unique $v$ in $C$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$
We say that $v$ is the projection of $u$ onto $C$ and we denote $d(u, C) = \|u - v\|$.

Let $p$ and $q$ be two strictly positive reals such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that, for all non-negative reals $a$ and $b$,
$$ab \leqslant \frac{a^{p}}{p} + \frac{b^{q}}{q}$$
You may use the concavity of the logarithm.

Let $p$ and $q$ be two strictly positive reals such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that, for all non-negative reals $a$ and $b$,
$$ab \leqslant \frac{a^{p}}{p} + \frac{b^{q}}{q}$$
You may use the concavity of the logarithm.

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
We decide to replace all $\rho_{k}$ by zeros. We thus replace the corresponding factors of $R_{N}$, $$\frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}},$$ by factors $X^{2}$. We thus obtain a new polynomial $S_{N}$ of the same degree as $R_{N}$.
Show that $0 \leqslant S_{N}(x) \leqslant R_{N}(x)$ for all $x \in [-1,1]$, then that $S_{N} \in B_{N}$.

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$ After replacing all $\rho_k$ by zeros we obtained $S_N$.
Similarly, in the list of $c_{j}$, we decide to replace those that do not belong to $[-1,1]$ by zeros. We thus replace the corresponding factors of $S_{N}$, $$\frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}}$$ by factors $X^{2}$. We thus obtain a new polynomial $T_{N}$.
Show that $0 \leqslant T_{N}(x) \leqslant S_{N}(x)$ for all $x \in [-1,1]$, then that $T_{N} \in B_{N}$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Deduce that $\forall x \in U, f(x) \leqslant \sup_{y \in \partial U} f(y)$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f_1$ and $f_2$ be two functions continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. Show that if the functions $f_1$ and $f_2$ are equal on $\partial U$, then $f_1$ and $f_2$ are equal on $U$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f_1$ and $f_2$ be two functions continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. Show that if the functions $f_1$ and $f_2$ are equal on $\partial U$, then $f_1$ and $f_2$ are equal on $U$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the Frobenius norm $\|\cdot\|_{F}$. We fix a unit vector $u$ in $\mathbb{R}^{d}$, define $g(M) = \|M \cdot u\|$, and let $C = \{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Let $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. Show that $P$ is invertible and that $P^{-1} \in \bigoplus_{p=0}^{n-1} \Delta_{p(k+1)}$.