Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. Suppose in this question that the functions $f, g_1, \ldots, g_p$ are convex. Let $x^{\star} \in K$ and $\mu_1^{\star}, \ldots, \mu_p^{\star} \in \mathbb{R}_+$ be such that $$\left\{ \begin{array}{l} \nabla f(x^{\star}) + \sum_{i=1}^{p} \mu_i^{\star} \nabla g_i(x^{\star}) = 0 \\ \mu_i^{\star} g_i(x^{\star}) = 0 \text{ for all } i \in \llbracket 1, p \rrbracket \end{array} \right.$$ is verified. Show that $f$ admits at $x^{\star}$ a global minimum on $K$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. Show that $\inf_{x \in K} f(x) = \inf_{x \in \mathbb{R}^n} \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(x, \mu)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. Show that for all $\mu \in \mathbb{R}_+^p$, there exists a unique $x_\mu \in \mathbb{R}^n$ satisfying $\mathcal{L}(x_\mu, \mu) = \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$. We say that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$ if $$\mathcal{L}(\bar{x}, \bar{\mu}) = \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \bar{\mu}) \quad \text{and} \quad \mathcal{L}(\bar{x}, \bar{\mu}) = \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(\bar{x}, \mu).$$ We assume in this question that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$. a. Show that $\bar{x}$ is a solution of $(P)$: $\inf_{x \in K} f(x)$. b. Show that $\bar{\mu}$ is a solution of $(Q)$: $\sup_{\mu \in \mathbb{R}_+^p} G(\mu)$. c. Show that $\inf_{x \in \mathbb{R}^n} \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(x, \mu) = \sup_{\mu \in \mathbb{R}_+^p} \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$. We consider $x^{\star} \in K$ a solution of $(P)$ satisfying hypothesis $(H)$. Let $\mu^{\star} = (\mu_1^{\star}, \ldots, \mu_p^{\star})$ as in question III.7. Show that $\mu^{\star}$ is a solution of $(Q)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$. We assume throughout this question that the function $\mu \in \mathbb{R}_+^p \mapsto x_\mu$ is continuous. We consider a solution $\bar{\mu} \in \mathbb{R}_+^p$ of $(Q)$. a. Let $\mu \in \mathbb{R}_+^p$ and $\xi \in \mathbb{R}^p$ be such that $\mu + \xi \in \mathbb{R}_+^p$. Show that for all $t \in [0,1]$, $\mu + t\xi \in \mathbb{R}_+^p$, and $$\lim_{\substack{t \rightarrow 0 \\ t > 0}} \frac{G(\mu + t\xi) - G(\mu)}{t} = \langle g(x_\mu), \xi \rangle.$$ Deduce that for all $\mu \in \mathbb{R}_+^p$, $\langle g(x_{\bar{\mu}}), \mu - \bar{\mu} \rangle \leqslant 0$. b. Show that $x_{\bar{\mu}}$ is a solution of $(P)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$. (Uzawa's Theorem). Let $A \in \mathcal{M}_{p,n}(\mathbb{R})$ be a matrix of rank $p$ and $b \in \mathbb{R}^p$. We assume that the function $g$ is of the form $$g : x \mapsto Ax + b.$$ a. Show that for all $\mu \in \mathbb{R}_+^p$, $\nabla f(x_\mu) = -{}^t A \mu$, and deduce that the function $\mu \mapsto x_\mu$ is continuous on $\mathbb{R}_+^p$. b. Show that $(P)$ admits a unique solution $x^{\star} \in K$, and that $(Q)$ admits a unique solution $\mu^{\star} \in \mathbb{R}_+^p$. Let $\rho > 0$. We define the sequence $(\mu^k)_{k \in \mathbb{N}}$ by recursion as follows:

• we fix $\mu^0 \in \mathbb{R}_+^p$,
• for all $k \in \mathbb{N}$, we set $\mu^{k+1} = P_{\mathbb{R}_+^p}\left(\mu^k + \rho g(x_{\mu^k})\right)$,

where $P_{\mathbb{R}_+^p} : \mathbb{R}^p \rightarrow \mathbb{R}_+^p$ denotes the projection onto the closed convex set $\mathbb{R}_+^p$ of $\mathbb{R}^p$. c. Show that $\mu^{\star} = P_{\mathbb{R}_+^p}\left(\mu^{\star} + \rho g(x_{\mu^{\star}})\right)$. We now assume that $\|Ax\| \leqslant \sqrt{\frac{\alpha}{\rho}} \|x\|$ for all $x \in \mathbb{R}^n$. d. Show that the sequence $(x_{\mu^k})_{k \in \mathbb{N}}$ converges to $x^{\star}$. e. Show that the sequence $(\mu^k)_{k \in \mathbb{N}}$ converges to $\mu^{\star}$.

1. a. Verify that the relative extrema of functions in $S$ are strict. b. Let $f \in S$. Show that the restriction of $f$ to the closure of each component of $\mathbb{R} \backslash E(f)$ is strictly monotone. Deduce that if $x \in E(f) \backslash \{\operatorname{Max} E(f)\}$ is a relative maximum (resp. minimum), the smallest element $y$ of $E(f)$ satisfying $y > x$ is a relative minimum (resp. maximum). c. Let $f \in S_n$ with $n \geq 2$. We set $\mathscr{E}(f) = f(E(f))$. Let $\sigma_f$ be the element of $\Sigma_n$ defined by $$\sigma_f = \beta_{\mathscr{E}(f)} \circ f \circ \beta_{E(f)}^{-1}$$ Show that $\sigma_f \in \operatorname{MD}(n) \cup \operatorname{DM}(n)$.

2. We define a relation $\sim$ on $S$ as follows: for every pair $(f, g)$ of $S^2$, $f \sim g$ if and only if there exist two continuous bijections $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ and $\psi : \mathbb{R} \rightarrow \mathbb{R}$, strictly increasing, which satisfy $f = \psi \circ g \circ \varphi$. a. Verify that $\sim$ is an equivalence relation on $S$ and show that each equivalence class of $\sim$ is contained in one of the sets $S_n, n \in \mathbb{N}$. b. Let $n \in \mathbb{N}^*$ and $\{u_1, \ldots, u_n\}, \{v_1, \ldots, v_n\}$ be subsets of $\mathbb{R}$ satisfying $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Verify that there exists a continuous bijection $\chi : \mathbb{R} \rightarrow \mathbb{R}$ strictly increasing such that $\chi(u_k) = v_k$ for $1 \leq k \leq n$. c. Suppose that $f$ and $g$ are in $S_*$ and that $$\lim_{x \rightarrow \pm\infty} |f(x)| = +\infty, \quad \lim_{x \rightarrow \pm\infty} |g(x)| = +\infty$$ Prove that $f \sim g$ if and only if $\sigma_f = \sigma_g$. d. Does the preceding equivalence hold for two arbitrary functions $f$ and $g$ of $S_*$?

3. We denote $C_b^0$ the space of continuous bounded functions from $\mathbb{R}$ to $\mathbb{R}$, equipped with the uniform norm: $\|f\| = \operatorname{Sup}_{x \in \mathbb{R}} |f(x)|$ for $f \in C_b^0$. a. Let $n \in \mathbb{N}^*, \{u_1, \ldots, u_n\} \subset \mathbb{R}$ and $\{v_1, \ldots, v_n\} \subset \mathbb{R}$ with $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Show that there exists a continuous map $\zeta : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ such that:

• for $s \in [0,1]$, the function $x \mapsto \zeta(s, x)$ is a strictly increasing bijection from $\mathbb{R}$ to $\mathbb{R}$,
• $\zeta(0, x) = x$ for $x \in \mathbb{R}$ and $\zeta(1, u_k) = v_k$, $1 \leq k \leq n$.

b. Prove that the equivalence classes of the restriction of $\sim$ to $S_* \cap C_b^0$ are arc-connected. c. Give an example of a continuous arc $\gamma : [0,1] \rightarrow S \cap C_b^0$ such that $\gamma(0) \in S_0$ and $\gamma(1) \in S_2$.

1. Let $n \in \mathbb{N}^*$. We denote Id the identity map of $\mathbb{R}^n$. We equip $\mathbb{R}^n$ with a norm denoted $\|\cdot\|$ and the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the associated norm, also denoted $\|\cdot\|$. For $x \in \mathbb{R}^n$ and $r \in \mathbb{R}^+$, we denote $B(x, r)$ (resp. $B(x, r]$) the open (resp. closed) ball with center $x$ and radius $r$. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ containing 0 and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ such that $f(0) = 0$ and whose differential $\varphi$ at 0 is invertible. a. We set $g = \mathrm{Id} - \varphi^{-1} \circ f$. Show that $g$ is of class $C^1$ on $\mathcal{O}$ and that there exists $\varepsilon > 0$ such that $B(0, \varepsilon) \subset \mathcal{O}$ and $\|Dg(x)\| \leq \frac{1}{2}$ for $x \in B(0, \varepsilon)$. Deduce that $f$ is injective in $B(0, \varepsilon)$. b. Let $0 < r < \varepsilon$ and let $z_0 \in B(0, r/2)$. We set $h(x) = g(x) + z_0$ for $x \in \mathcal{O}$. Show that $$h(B(0, r]) \subset B(0, r].$$ c. Show that there exists $a \in B(0, r]$ such that $f(a) = \varphi(z_0)$. d. Let $W = \varphi(B(0, r/2))$ and $V = f^{-1}(W) \cap B(0, \varepsilon)$. Show that $V$ and $W$ are open and that $f_{|V}$ is a homeomorphism from $V$ to $W$.

2. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ whose differential at $x$ is invertible for all $x \in \mathcal{O}$. Prove that the image by $f$ of an open set of $\mathcal{O}$ is an open set of $\mathbb{R}^n$.

3. For $n \geq 2$, let $O_{n-1} = \{(x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n-1} \mid 0 < x_1 < x_2 < \cdots < x_{n-1}\}$ and let $U_{n-1}$ be the set of $(n-1)$-tuples $(y_1, \ldots, y_{n-1}) \in \mathbb{R}^{n-1}$ such that $$0 < y_1, \quad y_i > y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is odd}, \quad y_i < y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is even}.$$ For $x \in O_{n-1}$, we define the function $\pi_x \in \mathscr{P}_n$ by $\pi_x(t) = t(x_1 - t) \cdots (x_{n-1} - t)$. We define the map $Y = (Y_1, \ldots, Y_{n-1}) : O_{n-1} \rightarrow \mathbb{R}^{n-1}$ by $$Y_i(x) = \int_0^{x_i} \pi_x(u)\, du, \quad x = (x_1, \ldots, x_{n-1}) \in O_{n-1}$$ a. Let $j \in \{1, \ldots, n-1\}$ and $x \in O_{n-1}$. Show that $$d_{x,j} : t \mapsto \int_0^t u \prod_{1 \leq \ell \leq n-1, \ell \neq j} (x_\ell - u)\, du$$ is in $\mathscr{P}_n$ and vanishes with its derivative at 0. Deduce the existence of $\chi_{x,j} \in \mathscr{P}_{n-2}$ satisfying $$\forall t \in \mathbb{R}, \quad d_{x,j}(t) = t^2 \chi_{x,j}(t).$$ b. For $x \in O_{n-1}$ and $(i,j) \in \{1, \ldots, n-1\}^2$, show the existence of $\frac{\partial Y_i}{\partial x_j}(x)$ and verify that $$\frac{\partial Y_i}{\partial x_j}(x) = d_{x,j}(x_i)$$ Deduce that $Y$ is a map of class $C^1$ on the open set $O_{n-1}$, with values in $U_{n-1}$. c. Prove that for $x \in O_{n-1}$, the set $\{\chi_{x,j} \mid j \in \{1, \ldots, n-1\}\}$ is a basis of $\mathscr{P}_{n-2}$. d. Deduce that the differential of $Y$ at point $x$ is invertible.

4. For $n \in \mathbb{N}$, a function of $\mathscr{P}_n$ is said to be monic when the coefficient of its term of degree $n$ is 1. We denote $\mathscr{P}_n^u$ the set of these functions. We denote $C_n = \operatorname{Inf}\left\{\int_0^1 |f(t)|\, dt \mid f \in \mathscr{P}_n^u\right\}$. a. Show that $C_n > 0$. b. For $n \geq 2$, prove that if $x \in O_{n-1}$ $$(x_{n-1})^{n+1} \leq \frac{1}{C_n}\left[Y_1(x) + \sum_{i=1}^{n-2} (-1)^i (Y_{i+1}(x) - Y_i(x))\right]$$ c. Verify that the map $Y$ extends continuously to the closure of $O_{n-1}$. d. Show that if $K$ is a compact subset of $\mathbb{R}^{n-1}$ contained in $U_{n-1}$, $Y^{-1}(K)$ is compact.

5. Show that $Y(O_{n-1})$ is open and closed in $U_{n-1}$ and deduce that $Y$ is surjective.

Can there be equality in the preceding inequality? $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$

Let $n$ be a non-zero natural number. Can there be equality in the inequality $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}?$$

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Express the function $$x \mapsto \begin{cases} \frac{\ln(1+x)}{x} & \text{if } x \in ]-1,1[ \setminus \{0\} \\ 1 & \text{if } x = 0 \end{cases}$$ using a Gauss hypergeometric function.

Let $f \in L^1(\mathbb{R})$. The Fourier transform of $f$ is defined by $$\forall \xi \in \mathbb{R}, \quad \hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \mathrm{e}^{-\mathrm{i}x\xi} \,\mathrm{d}x$$
Show that, for every function $f \in L^1(\mathbb{R})$, $\hat{f}$ is defined and continuous on $\mathbb{R}$.

Let $f \in L^1(\mathbb{R})$, where the Fourier transform of $f$ is defined by $$\forall \xi \in \mathbb{R}, \quad \hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \mathrm{e}^{-\mathrm{i}x\xi} \,\mathrm{d}x$$ Show that, for every function $f \in L^1(\mathbb{R})$, $\hat{f}$ is defined and continuous on $\mathbb{R}$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ We admit that, in case of existence of all quantities present in the following expression, $$F_{a,b,c}(1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}.$$ Let $N \in \mathbb{N}, c \in D, a \in \mathbb{R}$ such that $c - a \in D$. Justify the existence of $F_{a,-N,c}(1)$ and prove that $$\sum_{k=0}^{N} (-1)^k \binom{N}{k} \frac{[a]_k}{[c]_k} = \frac{[c-a]_N}{[c]_N}.$$

Show that the map $f \mapsto \hat{f}$ is a continuous linear map from the normed vector space $(L^1(\mathbb{R}), \|\cdot\|_1)$ to the normed vector space $(L^\infty(\mathbb{R}), \|\cdot\|_\infty)$.

Show that the map $f \mapsto \hat{f}$ is a continuous linear map from the normed vector space $(L^1(\mathbb{R}), \|\cdot\|_1)$ to the normed vector space $(L^\infty(\mathbb{R}), \|\cdot\|_\infty)$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ We admit that, in case of existence of all quantities present in the following expression, $$F_{a,b,c}(1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}.$$ Let $(u, v) \in \mathbb{N}^2$ such that $N \leqslant \min(u, v)$. By taking $a = -u$ and $c = v - N + 1$, show Vandermonde's identity: $$\binom{u+v}{N} = \sum_{k=0}^{N} \binom{u}{k} \binom{v}{N-k}.$$

Let $f \in L^1(\mathbb{R})$, $\lambda \in \mathbb{R}_+^*$ and let $g$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that $g(x) = f(\lambda x)$ for all real $x$. Show that $g \in L^1(\mathbb{R})$ and, for all real $\xi$, express $\hat{g}(\xi)$ in terms of $\hat{f}$, $\xi$ and $\lambda$.