Let $f$ be a non-constant entire function with $f(z) \neq 0$ for all $z \in \mathbb{C}$. Consider the set $U = \{z : |f(z)| < 1\}$. Show that all connected components of $U$ are unbounded.

1. Given sets $A = \{ 1,2,3 \} , B = \{ 2,3 \}$, then
A. $\mathrm { A } = \mathrm { B }$
B. $\mathrm { A } \cap \mathrm { B } = \varnothing$
C. $A \subset B$
D. $B \subset A$

If $A$ is a subset of $E$, we denote $A^{\perp\varphi} = \{ x \in E \mid \forall a \in A,\ \varphi(x,a) = 0 \}$. Show that $A^{\perp\varphi}$ is a vector subspace of $E$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We say that two vector subspaces $F$ and $G$ of $E$ are orthogonal if and only if for all $(x,y) \in F \times G$, $\varphi(x,y) = 0$.
If $F$ and $G$ are two vector subspaces of $E$ that are orthogonal and non-singular, show that $F \oplus G$ is non-singular.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that $\bar{F} = G \oplus P_1 \oplus \ldots \oplus P_s$ is non-singular. We will say that $\bar{F}$ is a non-singular completion of $F$.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. If $V$ and $V ^ { \prime }$ are two $\mathbb { K }$-vector spaces of finite dimension, $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ are two non-degenerate quadratic forms, the orthogonal sum $q \perp q ^ { \prime }$ of $q$ and $q ^ { \prime }$ is the quadratic form on $V \times V ^ { \prime }$ defined by $$q \perp q ^ { \prime } \left( x , x ^ { \prime } \right) = q ( x ) + q ^ { \prime } \left( x ^ { \prime } \right)$$ for all $x \in V$ and all $x ^ { \prime } \in V ^ { \prime }$.
Let $V , V ^ { \prime }$ and $V ^ { \prime \prime }$ be three $\mathbb { K }$-vector spaces of finite dimension and $\left( q , q ^ { \prime } , q ^ { \prime \prime } \right) \in \mathcal { Q } ( V ) \times \mathcal { Q } \left( V ^ { \prime } \right) \times \mathcal { Q } \left( V ^ { \prime \prime } \right)$.
(a) Show that $q \perp q ^ { \prime } \in \mathcal { Q } \left( V \times V ^ { \prime } \right)$ and then that $\left( q \perp q ^ { \prime } \right) \perp q ^ { \prime \prime } \cong q \perp \left( q ^ { \prime } \perp q ^ { \prime \prime } \right)$.
(b) Show that if $q ^ { \prime } \cong q ^ { \prime \prime }$ then $q \perp q ^ { \prime } \cong q \perp q ^ { \prime \prime }$.
(c) Prove that if $V = V ^ { \prime } \oplus V ^ { \prime \prime }$ and $\tilde { q } ( x , y ) = 0$ for all $x \in V ^ { \prime }$ and all $y \in V ^ { \prime \prime }$, then $q \cong q ^ { \prime } \perp q ^ { \prime \prime }$ where $q ^ { \prime }$ is the restriction of $q$ to $V ^ { \prime }$ and $q ^ { \prime \prime }$ is the restriction of $q$ to $V ^ { \prime \prime }$.

Let $\Omega$ be a non-empty open set of $\mathbb{R}^2$ and $P$ a polynomial of two variables, such that $P(x,y) = 0$ for all $(x,y) \in \Omega$.
a) Show that for all $(x,y) \in \Omega$, the open set $\Omega$ contains a subset of the form $I \times J$, where $I$ and $J$ are non-empty open intervals of $\mathbb{R}$ containing $x$ and $y$ respectively.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Deduce that $P$ is the zero polynomial.
One may reduce to studying polynomial functions of one variable.

Let $\mathcal{U}$ and $\mathcal{V}$ be two vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} > n.$$ Show that $\mathcal{U} \cap \mathcal{V}$ is not reduced to $\{0\}$.

Let $\mathcal{K}$ be a compact convex set in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{K}}$.
9a. Show that the set of $\lambda \geqslant 0$ such that $-\lambda \mathcal{K} \subset \mathcal{K}$ is an interval.
We denote $$a(\mathcal{K}) = \sup\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$$
9b. Show that $a(\mathcal{K}) < \infty$ and that $a(\mathcal{K}) = \max\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$.
9c. Show that $0 < a(\mathcal{K}) \leqslant 1$. Deduce that $a(\mathcal{K}) = 1$ if and only if $\mathcal{K}$ is symmetric with respect to 0.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that the family $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ is a basis of $\mathbb{R}_n[X]$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$. Let $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$ and denote $d = b - a$. Let $k, n \in \mathbb{N}$ such that $k \leqslant n-1$. Show that $$\Lambda \cap [na + kd,\, na + (k+1)d] = \{na + kd,\, na + (k+1)d\}$$

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
Show that one of the two following assertions is true: (i) $\operatorname{ker}(T) \subset F^+$, (ii) $\operatorname{ker}(T) \subset F^-$.

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 $\mathcal{C}^2(U, \mathbb{R})$.

Show that if the union of a finite number of vector subspaces $F_1, \ldots, F_r$ of $E$ is a vector subspace, then one of the vector subspaces $F_i$ contains all the others.

Show that if the union of a finite number of vector subspaces $F_1, \ldots, F_r$ of $E$ is a vector subspace, then one of the vector subspaces $F_i$ contains all the others.

In $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$, let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system (a sequence of polynomials that is orthogonal and where each $V_n$ is monic of degree $n$). Show that, for all $n \in \mathbb { N }$, the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ is an orthogonal basis of the vector space $\mathbb { R } _ { n } [ X ]$ of polynomials with real coefficients of degree at most $n$.

Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and the polynomials $L_0, \ldots, L_n$ defined in Q5.
Show that $(L_0, \ldots, L_n)$ is a basis of $\mathbb{R}_n[X]$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that $\|P\|_K$ belongs to $\mathbb{R}$ for all $P \in \mathbb{C}[X]$.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
Prove that there exists $k \in \llbracket 1; n \rrbracket$ such that $v_{k} \notin F$ and that then $F$ and the vector line spanned by $v_{k}$ are in direct sum.

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$. Let $I \in \mathbb{N}^*, x_1, \ldots, x_I \in A^I$ and $(\lambda_1, \ldots, \lambda_I) \in \mathbb{R}_+^I$ such that $\sum_{i=1}^I \lambda_i = 1$, show that:

• a) $\sum_{i=1}^I \lambda_i x_i \in A$,
• b) if $x := \sum_{i=1}^I \lambda_i x_i \in \operatorname{Ext}(A)$ then $x_i = x$ for all $i \in \{1, \ldots, I\}$ such that $\lambda_i > 0$.

Let $E$ be a subset of $\mathbb{R}^d$. Recall that $$\operatorname{co}(E) := \left\{\sum_{i=1}^I \lambda_i x_i, I \in \mathbb{N}^*, \lambda_i \geq 0, \sum_{i=1}^I \lambda_i = 1, (x_1, \ldots, x_I) \in E^I\right\}.$$ Show that $\operatorname{co}(E)$ is the smallest convex set containing $E$ and that $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. The polar cone of $E$ is defined by $$E^+ := \left\{p \in \mathbb{R}^d : p \cdot x \geqslant 0, \forall x \in E\right\}$$ and its bi-polar cone by $$E^{++} = (E^+)^+ := \left\{\xi \in \mathbb{R}^d : \xi \cdot p \geqslant 0, \forall p \in E^+\right\}.$$ Show that $E^+$ and $E^{++}$ are closed convex cones and that $E \subset E^{++}$.

Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and $$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$ show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:

• $\xi \in F$,
• $\xi \cdot x \geqslant 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geqslant 0, i = 1, \ldots, k$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Denote by $C$ the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Show that $C$ is non-empty, convex, closed and bounded.