Let $f : \mathbb { R } \longrightarrow ( 0 , \infty )$ be an infinitely differentiable function with $\int _ { - \infty } ^ { \infty } f ( t ) d t = 1$. Pick the correct statement(s) from below.
(A) $f ( t )$ is bounded.
(B) $\lim _ { | t | \rightarrow \infty } f ^ { \prime } ( t ) = 0$.
(C) There exists $t _ { 0 } \in \mathbb { R }$ such that $f \left( t _ { 0 } \right) \geq f ( t )$ for all $t \in \mathbb { R }$.
(D) $f ^ { \prime \prime } ( a ) = 0$ for some $a \in \mathbb { R }$.

Pick the correct statement(s) from below.
(A) If $f$ is continuous and bounded on $( 0,1 )$, then $f$ is uniformly continuous on $( 0,1 )$.
(B) If $f$ is uniformly continuous on $( 0,1 )$, then $f$ is bounded on $( 0,1 )$.
(C) If $f$ is continuous on $( 0,1 )$ and $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ and $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ exists, then $f$ is uniformly continuous on $( 0,1 )$.
(D) Product of a continuous and a uniformly continuous function on $[ 0,1 ]$ is uniformly continuous.

Let $X$ be the metric space of real-valued continuous functions on the interval $[ 0,1 ]$ with the ``supremum distance'': $$d ( f , g ) = \sup \{ | f ( x ) - g ( x ) | : x \in [ 0,1 ] \} \text { for all } f , g \in X$$ Let $Y = \{ f \in X : f ( [ 0,1 ] ) \subset [ 0,1 ] \}$ and $Z = \left\{ f \in X : f ( [ 0,1 ] ) \subset \left[ 0 , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , 1 \right] \right\}$. Pick the correct statement(s) from below.
(A) $Y$ is compact.
(B) $X$ and $Y$ are connected.
(C) $Z$ is not compact.
(D) $Z$ is path-connected.

Let $X : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z \leq 0 \right.$, or $\left. x , y \in \mathbb { Q } \right\}$ with subspace topology. Pick the correct statement(s) from below.
(A) $X$ is not locally connected but path connected.
(B) There exists a surjective continuous function $X \longrightarrow \mathbb { Q } _ { \geq 0 }$ (the set of non-negative rational numbers, with the subspace topology of $\mathbb { R }$ ).
(C) Let $S$ be the set of all points $p \in X$ having a compact neighbourhood (i.e. there exists a compact $K \subset X$ containing $p$ in its interior). Then $S$ is open.
(D) The closed and bounded subsets of $X$ are compact.

Let $Q$ be the space of infinite sequences $$\mathbf { x } : = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots \right)$$ of real numbers $x _ { n } \in [ 0,1 ]$, with the product topology coming from the identification $Q = [ 0,1 ] ^ { \mathbb { N } }$. ($[ 0,1 ]$ has the euclidean topology.) Let $S : Q \longrightarrow \mathbb { R }$ be the map $$S ( \mathbf { x } ) : = \sum _ { n } \frac { x _ { n } } { n ^ { 2 } } .$$ (A) Let $Q _ { 2 } : = \left\{ \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \left\lvert \, 0 \leq y _ { n } \leq \frac { 1 } { n } \right. \right\}$. Show that $Q _ { 2 }$ is compact.
(B) Let $D : Q _ { 2 } \longrightarrow Q$ be the map $$\left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \mapsto \left( y _ { 1 } , 2 y _ { 2 } , \ldots , n y _ { n } , \ldots \right)$$ Show that $D$ is a homeomorphism. (Hint: first show that $Q$ is Hausdorff.)
(C) Show that $S \circ D : Q _ { 2 } \longrightarrow \mathbb { R }$ is continuous. (Hint: Show that there is a suitable inner-product space $( L , \langle - , - \rangle )$ and a vector $\mathbf { a } \in L$ such that $( S \circ D ) ( \mathbf { x } ) = \langle \mathbf { x } , \mathbf { a } \rangle$ for each $\mathbf { x } \in Q _ { 2 }$.)
(D) Show that $S$ is continuous.

Let $f ( x ) = \left| \frac { \sin x } { x } \right| ^ { 1.001 }$ for $x \neq 0$ and $f ( 0 ) = L$ such that $f$ is continuous. Let $I ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
Statements
(21) $L = 1.001$ (22) $I ( 0.001 ) > 0.001$. (23) As $x \rightarrow \infty$ the limit of $I ( x )$ is greater than 1001 (possibly $\infty$). (24) The function $I ( x )$ is NOT differentiable at infinitely many points.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Establish that $f$ is in $\mathcal{B}_1$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Is the function $\frac{\partial f}{\partial x}$ in $\mathcal{B}_2$?

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Justify the convergence, for any real $q \geqslant 0$, of $\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2 - q^2}}\,\mathrm{d}r$.

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Verify that $\hat{f}$ is defined on $\mathbb{R}^2$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that $\bar{f}$ is of class $C^1$ on $\mathbb{R}$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that the function $r \mapsto r^2 \bar{f}(r)$ is bounded on $\mathbb{R}$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Show that if we further assume that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$, then $r \mapsto r^4 \bar{f}^{\prime}(r)$ is bounded on $\mathbb{R}$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that $H$ is continuous on $]0, +\infty[$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that near $+\infty$ we have $H(q) = O\left(\frac{1}{q^2}\right)$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Prove that if we further assume that $r \mapsto r^4 h^{\prime}(r)$ is bounded, then the function $H$ is of class $C^1$ on $]0, +\infty[$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set, with the notations of part III: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Justify that $F$ is of class $C^1$ on $]0, +\infty[$ and that near $+\infty$ we have $F(q) = O\left(\frac{1}{q}\right)$.

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real: $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
In the rest of this part, the pair $(x,y)$ is fixed in $D(0,1)$.
Show that $t \mapsto \mathrm{N}(x,y,t)$ is defined and continuous on $[0, 2\pi]$.

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$\mathrm{N}_f(x,y) = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t) f(\cos t, \sin t)\, \mathrm{d}t$$ on $D(0,1)$.
In this question, we fix $t_0 \in [0,2\pi]$, $(x,y) \in D(0,1)$ and $\varepsilon > 0$. Moreover, we denote, for all real $\delta > 0$: $$I_0^\delta = \left\{ t \in [0,2\pi] \mid \|(\cos t, \sin t) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta \right\}$$
a) Show that $I_0^\delta$ is an interval or the union of two disjoint intervals.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Show, using the application $f$, the existence of a real $\delta > 0$ such that $$\left| \int_{t \in I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$
c) Let $\delta > 0$ be arbitrary. Show that, if $t \in [0,2\pi] \backslash I_0^\delta$ and $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta/2$, then $$|\mathrm{N}(x,y,t)| \leqslant 4 \frac{1 - (x^2 + y^2)}{\delta^2}$$
d) Deduce from the previous question that, for $\delta > 0$ fixed, there exists $\eta > 0$ such that, if $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \eta$, then $$\left| \int_{t \in [0,2\pi] \backslash I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$

We consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$ for all $x \in \mathbb{R}$.
Show that there exists a real number $M > 0$ such that $f$ is an $M$-Lipschitz function.

Show that $g_{\sigma}$ is integrable on $\mathbb{R}$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. Show that, for any real $\xi$, the function $\left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto f(x) \exp(-\mathrm{i} 2\pi \xi x) \end{aligned}\right.$ is integrable on $\mathbb{R}$.

We denote by $\mathbb{R}_{N}[X]$ the vector space of polynomials with real coefficients, of degree at most $N$. We define the set $A_{N}$ formed by $P \in \mathbb{R}_{N}[X]$, such that $P(-1) = P(1) = 1$, which furthermore satisfy $P(x) \geqslant 0$ for all $x$ in the interval $[-1,1]$. We define on $\mathbb{R}_{N}[X]$ a linear form $L$ by $L(P) = \int_{-1}^{1} P(x)\,dx$. The infimum of $L$ on $A_N$ is denoted $a_N = \inf\{L(P) \mid P \in A_N\}$.
(a) Show that the infimum of $L$ on $A_{N}$ is attained.
In what follows, we denote by $B_{N}$ the set of $P \in A_{N}$ such that $L(P) = a_{N}$.
(b) Show that $B_{N}$ is a convex compact subset.
(c) Verify that $B_{N}$ contains an even polynomial.

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 $u = (u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Show that $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$ is a convex and closed subset of $\mathcal{M}_{k,d}(\mathbb{R})$.

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
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Show that $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$ is a convex and closed subset of $\mathcal{M}_{k,d}(\mathbb{R})$.