After justifying the existence of the suprema, show that: $$\sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| } = \sup _ { \substack { x \in E \\ \| x \| = 1 } } \| u ( x ) \| .$$

To every $p \in \mathbb{R}[X]$, we associate the function $J(p) = Jp$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad J(p)(x) = Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$$
Show that $J$ preserves degree and that $J$ is invertible.

We denote by $E$ the vector space of functions with real values continuous on $\mathbb { R } _ { + }$. For every element $f$ of $E$ and all $x \in \mathbb { R } _ { + }$ we set $F ( x ) = \int _ { 0 } ^ { x } f ( u ) \mathrm { d } u$.

1. Justify that $F$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$ and give for all $x \in \mathbb { R } _ { + }$ the expression of $F ^ { \prime } ( x )$.
   Let $\Psi : f \in E \mapsto \Psi ( f )$ defined by: $\forall x \in \mathbb { R } _ { + } , \Psi ( f ) ( x ) = \int _ { 0 } ^ { 1 } f ( x t ) \mathrm { d } t$.
2. Express, for all strictly positive real $x$, $\Psi ( f ) ( x )$ using $F ( x )$.
3. Justify that the function $\Psi ( f )$ is continuous on $\mathbb { R } _ { + }$ and give the value of $\Psi ( f ) ( 0 )$.
4. Show that $\Psi$ is an endomorphism of $E$.
5. Surjectivity of $\Psi$
   Let $h : x \in \mathbb { R } _ { + } \longmapsto h ( x ) = \left\{ \begin{array} { l l } x \sin \left( \frac { 1 } { x } \right) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$.
   1. [5.1.] Show that the function $h$ is continuous on $\mathbb { R } _ { + }$.
   2. [5.2.] Is the function $h$ of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$?
   3. [5.3.] Let $g \in \operatorname { Im } ( \Psi )$. Show that the function $x \mapsto x g ( x )$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$.
   4. [5.4.] Do we have $h \in \operatorname { Im } ( \Psi )$?
   5. [5.5.] Conclude.
6. Show that $\Psi$ is injective.
7. Search for the eigenvectors of $\Psi$
   1. [7.1.] Justify that 0 is not an eigenvalue of $\Psi$.
      Let $\mu \in \mathbb { R }$. We consider the differential equation $(L)$ on $\mathbb { R } _ { + } ^ { * }$: $$y ^ { \prime } + \frac { \mu } { x } y = 0$$
   2. [7.2.] Solve $(L)$ on $\mathbb { R } _ { + } ^ { * }$.
   3. [7.3.] Determine the solutions of $(L)$ that can be extended by continuity on $\mathbb { R } _ { + }$.
   4. [7.4.] Then determine the eigenvalues of $\Psi$ and the associated eigenspaces.
8. Let $n \in \mathbb { N } , n > 1$. For $i \in \llbracket 1 , n \rrbracket$, we set: $$f _ { i } : x \in \mathbb { R } _ { + } \longmapsto f _ { i } ( x ) = x ^ { i } \text { and } g _ { i } : x \in \mathbb { R } _ { + } \longmapsto g _ { i } ( x ) = \begin{cases} x ^ { i } \ln ( x ) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{cases}$$ We denote $\mathscr { B } = \left( f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \right)$ and $F _ { n }$ the vector subspace of $E$ generated by $\mathscr { B }$.
   1. [8.1.] We want to show that the family $\mathcal { B } = \left( f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \right)$ is a basis of $F _ { n }$.
      Let $\left( \alpha _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket }$ and $\left( \beta _ { j } \right) _ { j \in \llbracket 1 , n \rrbracket }$ be scalars such that $\sum _ { i = 1 } ^ { n } \alpha _ { i } f _ { i } + \sum _ { j = 1 } ^ { n } \beta _ { j } g _ { j } = 0$.
      1. [8.1.1.] Show that $\alpha _ { 1 } = \beta _ { 1 } = 0$. One may simplify the expression (*) by $x$ when $x$ is non-zero.
      2. [8.1.2.] Let $p \in \llbracket 1 , n - 1 \rrbracket$. Suppose that $\alpha _ { 1 } = \cdots = \alpha _ { p } = \beta _ { 1 } = \cdots = \beta _ { p } = 0$. Prove that $\alpha _ { p + 1 } = \beta _ { p + 1 } = 0$.
      3. [8.1.3.] Conclude and determine the dimension of the vector space $F _ { n }$.
   2. [8.2.] Where we prove that $\Psi$ induces an endomorphism on $F _ { n }$
      1. [8.2.1.] Let $x > 0$ and $p \in \mathbb { N } ^ { * }$. Show that the integral $\int _ { 0 } ^ { x } t ^ { p } \ln ( t ) \mathrm { d } t$ is convergent and calculate it.
      2. [8.2.2.] Deduce that $\Psi$ induces an endomorphism $\Psi _ { n }$ on $F _ { n }$.
   3. [8.3.] Give the matrix of the application $\Psi _ { n }$ in the basis $\mathcal { B }$.
   4. [8.4.] Prove that $\Psi _ { n }$ is an automorphism of $F _ { n }$.
   5. [8.5.] Let $z : x \in \mathbb { R } _ { + } \longmapsto z ( x ) = \left\{ \begin{array} { l l } \left( x + x ^ { 2 } \right) \ln ( x ) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$. After verifying that $z \in F _ { n }$, determine $\Psi _ { n } ^ { - 1 } ( z )$.

To every $p \in \mathbb{K}[X]$, we associate the function $L(p) = Lp$ from $\mathbb{K}$ to $\mathbb{K}$ defined by $$\forall x \in \mathbb{K}, \quad L(p)(x) = Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$$
Show that $\int_0^{+\infty} \mathrm{e}^{-t} t^k\,\mathrm{d}t$ exists for all $k \in \mathbb{N}$ and calculate its value.

Let $I$ be an interval of $\mathbf{R}$. Let $f : I \rightarrow \mathbf{R}$ be a convex function. Show that, for all $p \in \mathbf{N}^\star$, for all $(\lambda_1, \ldots, \lambda_p) \in (\mathbf{R}_+)^p$ such that $\sum_{i=1}^p \lambda_i = 1$ and for all $(x_1, \ldots, x_p) \in I^p$, we have: $$f\left(\sum_{i=1}^p \lambda_i x_i\right) \leq \sum_{i=1}^p \lambda_i f(x_i)$$ Hint: You may proceed by induction on $p$.

Let $I$ be an interval of $\mathbf { R }$. Let $f : I \rightarrow \mathbf { R }$ be a convex function. Show that, for all $p \in \mathbf { N } ^ { \star }$, for all $\left( \lambda _ { 1 } , \ldots , \lambda _ { p } \right) \in \left( \mathbf { R } _ { + } \right) ^ { p }$ such that $\sum _ { i = 1 } ^ { p } \lambda _ { i } = 1$ and for all $\left( x _ { 1 } , \ldots , x _ { p } \right) \in I ^ { p }$, we have:
$$f \left( \sum _ { i = 1 } ^ { p } \lambda _ { i } x _ { i } \right) \leq \sum _ { i = 1 } ^ { p } \lambda _ { i } f \left( x _ { i } \right)$$
Hint: You may proceed by induction on $p$.

Show that $S$ is a closed and path-connected subset of $\mathbb{H}$.

Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. Show the inequality $\frac{\operatorname{Tr}(M)}{n} \geq \operatorname{det}^{1/n}(M)$.
Hint: You may show that $x \mapsto -\ln(x)$ is convex on $\mathbf{R}_+^\star$.

Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. Show the inequality $\frac { \operatorname { Tr } ( M ) } { n } \geq \operatorname { det } ^ { 1 / n } ( M )$.
Hint: You may show that $x \mapsto - \ln ( x )$ is convex on $\mathbf { R } _ { + } ^ { \star }$.

Let $U, V \in \mathbb{H}^{\mathrm{im}}$. a) Show that $U$ and $V$ are orthogonal if and only if $UV + VU = 0$. In this case show that $UV \in \mathbb{H}^{\mathrm{im}}$ and that the determinant of the family $(U, V, UV)$ in the basis $(I, J, K)$ of $\mathbb{H}^{\mathrm{im}}$ is non-negative. b) Show that if $(U, V)$ is an orthonormal family in $\mathbb{H}^{\mathrm{im}}$, then $(U, V, UV)$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$.

Show that the set of shift-invariant endomorphisms of $\mathbb{K}[X]$ is a subalgebra of $\mathcal{L}(\mathbb{K}[X])$. Is the set of delta endomorphisms of $\mathbb{K}[X]$ closed under addition? under composition?

Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. You may use without proving it the inequality: $$\forall (x_1, \ldots, x_n) \in (\mathbf{R}_+)^n, \quad 2\max\{x_1, \ldots, x_n\}\left(\frac{1}{n}\sum_{k=1}^n x_k - \prod_{k=1}^n x_k^{1/n}\right) \geq \frac{1}{n}\sum_{k=1}^n \left(x_k - \prod_{j=1}^n x_j^{1/n}\right)^2.$$ Deduce that $$\frac{\operatorname{Tr}(M)}{n} - \operatorname{det}^{1/n}(M) \geq \frac{\left\|M - \operatorname{det}^{1/n}(M) I_n\right\|_2^2}{2n\|M\|_2}.$$

Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. In the rest of this part, you may use without proof the inequality below $\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbf { R } _ { + } \right) ^ { n }$,
$$2 \max \left\{ x _ { 1 } , \ldots , x _ { n } \right\} \left( \frac { 1 } { n } \sum _ { k = 1 } ^ { n } x _ { k } - \prod _ { k = 1 } ^ { n } x _ { k } ^ { 1 / n } \right) \geq \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \left( x _ { k } - \prod _ { j = 1 } ^ { n } x _ { j } ^ { 1 / n } \right) ^ { 2 }$$
Deduce that
$$\frac { \operatorname { Tr } ( M ) } { n } - \operatorname { det } ^ { 1 / n } ( M ) \geq \frac { \left\| M - \operatorname { det } ^ { 1 / n } ( M ) I _ { n } \right\| _ { 2 } ^ { 2 } } { 2 n \| M \| _ { 2 } }$$

Let $\left(a_k\right)_{k \in \mathbb{N}}$ be a sequence of elements of $\mathbb{K}$. For every polynomial $p \in \mathbb{K}[X]$, show that the expression $$\sum_{k=0}^{+\infty} a_k D^k p$$ makes sense and defines a polynomial of $\mathbb{K}[X]$.

Show that, for every sequence $\left(a_k\right)_{k \in \mathbb{N}}$ of elements of $\mathbb{K}$, $\sum_{k=0}^{+\infty} a_k D^k$ is a shift-invariant endomorphism.

Let $\left(a_k\right)_{k \in \mathbb{N}}$ and $\left(b_k\right)_{k \in \mathbb{N}}$ be sequences of elements of $\mathbb{K}$ such that $\sum_{k=0}^{+\infty} a_k D^k = \sum_{k=0}^{+\infty} b_k D^k$.
Show that, for all $k \in \mathbb{N}$, $a_k = b_k$.

Show the inequality $$\forall (A, B) \in S_n^{++}(\mathrm{R})^2, \quad \operatorname{det}^{1/n}(A + B) \geq \operatorname{det}^{1/n}(A) + \operatorname{det}^{1/n}(B)$$

Show the inequality
$$\forall ( A , B ) \in S _ { n } ^ { + + } ( \mathrm { R } ) ^ { 2 } , \quad \operatorname { det } ^ { 1 / n } ( A + B ) \geq \operatorname { det } ^ { 1 / n } ( A ) + \operatorname { det } ^ { 1 / n } ( B )$$

For every $n \in \mathbb{N}$, define the polynomial $q_n = \frac{X^n}{n!}$. Let $T$ be an endomorphism of $\mathbb{K}[X]$.
Show that $T$ is a shift-invariant endomorphism if, and only if, $$T = \sum_{k=0}^{+\infty} \left(T q_k\right)(0) D^k$$

Show that, if $A$ and $B$ belong to $S_n^{++}(\mathbf{R})$, then: $$\forall t \in [0,1], \quad \operatorname{det}((1-t)A + tB) \geq \operatorname{det}(A)^{1-t} \operatorname{det}(B)^t$$ Justify that this inequality remains valid for $A$ and $B$ only in $S_n^+(\mathbf{R})$.

Show that, if $A$ and $B$ belong to $S _ { n } ^ { + + } ( \mathrm { R } )$, then:
$$\forall t \in [ 0,1 ] , \quad \operatorname { det } ( ( 1 - t ) A + t B ) \geq \operatorname { det } ( A ) ^ { 1 - t } \operatorname { det } ( B ) ^ { t }$$
Justify that this inequality remains valid for $A$ and $B$ only in $S _ { n } ^ { + } ( \mathbf { R } )$.

Show that two shift-invariant endomorphisms of $\mathbb{K}[X]$ commute.

What can be deduced about the function $\ln \circ \det$ on $S_n^{++}(\mathrm{R})$?

What can we deduce about the function $\ln \circ \det$ on $S _ { n } ^ { + + } ( \mathbf { R } )$ ?

For all $p \in \mathbb{K}[X]$ non-zero and $a \in \mathbb{K}$, show, using question 11, that $$p(X+a) = \sum_{k=0}^{\deg(p)} \frac{a^k}{k!} p^{(k)}$$ where $p^{(k)}$ denotes the $k$-th derivative of the polynomial $p$. Recognize this formula.