\[
  \varphi(a) = \lim_{x \to 1} ax = \lim_{x \to 1} xa,
\]

\[
  \varphi\langle a, f\rangle = \lim_{x \to 1} \langle ax, f\rangle
\]

\[
  \langle v, f\rangle = \langle u, f_1\rangle
\]

\begin{align*}
  \text{(1)}\qquad & \mu_f = f(m) \quad\text{i.e.} \\
  \text{(2)}\qquad & \langle \alpha, \mu_f\rangle = \langle \alpha\circ f, m\rangle = \int \alpha\circ f\, dm
\end{align*}

\[
  \text{(3)}\qquad t = \int_s^\infty d\mu(s) \qquad (s = \varphi(t))\,.
\]

\[
  s \to \int_s^\infty d\mu(s)\,.
\]

\[
  \text{(4)}\qquad \varphi_f^{-}(t) = \inf_{\substack{E \subset M\\ m(E) \leq t}} \|(1-E)f\|_\infty \qquad
  \varphi_f^{+}(t) = \inf_{\substack{E \subset M\\ m(E) < t}} \|(1-E)f\|_\infty
\]

\[
  \text{(5)}\qquad \left\{
  \begin{array}{ll}
    \varphi_f \leq \varphi_g & \text{si } f \leq g \\
    \varphi_{\lambda f} = \lambda\varphi_f & \text{si } \lambda \geq 0 \\
    \varphi_{f^\alpha} = (\varphi_f)^\alpha & \text{si } \alpha \geq 0 \\
    \varphi_{f+g} \leq \varphi_f + \varphi_g & \\
    \varphi_{fg} \leq \varphi_f\varphi_g &
  \end{array}\right.
\]

\[
  \int_1^\infty (s-1)\, d\mu(s) < +\infty, \qquad\text{ou encore}\qquad
  \text{(6)}\quad \int^\infty s\, d\mu(s) < \struck{\ill{}} +\infty
\]

\[
  \text{(7)}\qquad \Phi_f(t) = \int_0^t \varphi_f(s)\, ds
\]

\[
  \text{(8)}\qquad \Psi_f(t) = \int_0^t \log\varphi_f(s)\, ds
\]

\[
  \text{(9)}\qquad \Delta_f(t) = \exp\Psi_f(t)
\]

\begin{align*}
  \text{(10)}\qquad & \Psi_{fg} \leq \Psi_f + \Psi_g \qquad\text{i.e.} \\
  \text{(11)}\qquad & \Delta_{fg} \leq \Delta_f\,\Delta_g
\end{align*}

\[
  \text{(12)}\qquad \Delta(f) = \Delta_{|f|}(|M|) = \struck{\exp} \int_M \log|f|\, dm
\]

\[
  \text{(13)}\qquad \Delta(fg) = \Delta(f)\,\Delta(g) \qquad \Delta(1) = 1
\]

\[
  \text{(14)}\qquad \varphi_{1+f}(t) = 1 + \varphi_f^{-}(t) \qquad \text{si } 0 < t \leq |M|
\]

\[
  \text{(15)}\qquad \Delta_{1+f}(t) = \exp\int_0^t \log(1 + \varphi_f(s))\, ds \qquad 0 \leq t \leq |M|
\]

\[
  \text{(16)}\qquad \Delta_{1+f}(|M|) = \Delta(1+f)
\]

\[
  \text{(17)}\qquad \int fg\, dm \leq \int_0^{|M|} \varphi_f\varphi_g\, ds
\]

\[
  W(\exp f) = \sup_i \Bigl( a_i + \int_0^{\omega} \varphi_i\,\varphi_f \Bigr) \qquad
  \uncertain{P}(f) = \sup_i \Bigl( a_i + \int_0^{\infty} \varphi_i \log\varphi_f \Bigr)
\]

\[
  W_\alpha(f) = \int_0^\omega \alpha(\varphi_f) \struck{\ill{}}
\]

\[
  \text{(1)}\qquad W_{\alpha_p}(f) = N_p(f)^p = \int |f|^p \struck{\ill{}}
\]

\[
  W(f) = \sup W(\varphi_g)
\]

\[
  P(f) = \sup_i \bigl( L_i(f) + a_i \bigr).
\]

\[
  P(f) \geq L_i(g) + a_i
\]

\[
  P(f) \geq \int \varphi_{L_i}\varphi_f + a_i, \qquad \text{Posons } \varphi_{L_i} = \varphi_i :
\]

\[
  \text{(1)}\qquad \boxed{\ P(f) = \sup_{i \in I}\ a_i + \int_0^{\omega} \varphi_i\,\varphi_f\ }
\]

\[
  \text{(2)}\qquad \boxed{\ P(f) = \sup_i\ a_i + \int_0^{\infty} \varphi_i\,\varphi_f\ }
\]

\[
  \text{(3)}\qquad P(f) = P^{*}(\varphi_f)
\]

\[
  \text{(4)}\qquad P^{*}(\varphi) = \sup_i \Bigl( a_i + \int_0^{\infty} \varphi_i\,\varphi \Bigr)
\]

\[
  \text{(1)}\qquad \boxed{\ \int_0^t f \leq \int_0^t g\ } \qquad\text{pour } 0 \leq t \leq \omega
\]

\[
  \text{(1')}\qquad P(f) \leq P(g)
\]

\[
  \int_0^\omega f\varphi \leq \int_0^\omega g\varphi
\]

\[
  \text{(3)}\qquad \int_0^\omega f\varphi = [\varphi F]_0^\omega + \int_0^\omega F(-d\varphi) = \varphi(\omega)F(\omega) + \int_0^\omega F(-d\varphi)
\]

\[
  \text{(4)}\qquad \Delta_f(t) \leq \Delta_g(t) < +\infty \qquad 0 \leq t \leq \omega
\]

\[
  \text{(5)}\qquad W(f) \leq W(g)\,.
\]

\[
  T(A+B) = T(A) + T(B) \quad\text{et}\quad T(\lambda A) = \lambda T(A)
\]

\[
  T(AB) = T(BA) \qquad A \in \underline{a},\ B \in \mathcal{A}
\]

\[
  T(\textstyle\sup_i A_i) = \sup_i T(A_i)
\]

\[
  \langle \varphi, \mu_f\rangle = \int_M \varphi(f) = T(\varphi(A))\,.
\]

\[
  \text{(1)}\qquad \langle \varphi, \mu_A\rangle = T(\varphi(A)) \qquad \varphi \in K(\mathbf{C}^{*})
\]

\[
  \text{(3)}\qquad \int z^n\, d\mu_A(z) = T(A^n) \qquad (n \geq 1)
\]

\[
  \text{(4)}\qquad \left\{
  \begin{array}{l}
    \Phi_A(t) = \displaystyle\int_0^t \varphi_A(s)\, ds \\[1ex]
    \Psi_A(t) = \displaystyle\int_0^t \log\varphi_A(s)\, ds \\[1ex]
    \Delta_A = \Delta_f = \exp\Psi_f \qquad \Delta_A(t) = \exp\displaystyle\int_0^t \log\varphi_{|A|}(s)\, ds
  \end{array}\right.
\]

\begin{align*}
  \text{(5)}\qquad & \Delta_A(t) = \Delta_{|A|}(t) = \Delta_{\varphi_{|A|}}(t) = \exp\int_0^t \log\varphi_{|A|}(s)\, ds \\
  \text{(6)}\qquad & \Delta(A) = \struck{\exp\Psi_A} \; \Delta_A(\omega) \qquad (\text{si } \omega = T(1) < +\infty)
\end{align*}

\[
  T(f) = \int_0^1 T(f(s))\, ds\,.
\]

\[
  \text{(5)}\qquad \varphi_A^{-}(t) = \inf_{\substack{E \in \mathcal{P}(\mathcal{A})\\ \operatorname{Tr} E < t}}\ \sup_{x \in (1-E)\mathcal{H}} (Ax, x)
  = \inf_{\substack{E \in \mathcal{P}(\mathcal{A})\\ \operatorname{Tr} E < t}} \|(1-E)A(1-E)\|
\]

\[
  \inf_{\substack{E \subset M\ \text{mesurable}\\ m(E) < t}} \|(1-E)A\|_\infty \geq \varphi_A^{-}(t)
\]

\[
  \text{(6)}\qquad \varphi_{|A|}^{-}(t) = \inf_{\substack{E \in \mathcal{P}(\mathcal{A})\\ T(E) < t}}\ \sup_{x \in (1-E)\mathcal{H}} \|Ax\|
  = \inf_{\substack{E \in \mathcal{P}(\mathcal{A})\\ T(E) < t}} \|A(1-E)\|
\]

\[
  \text{(7)}\qquad \left\{
  \begin{array}{l}
    \varphi_{|BA|} \leq \|B\|\, \varphi_{|A|} \\[0.5ex]
    \varphi_{|AB|} \leq \|B\|\, \varphi_{|A|}
  \end{array}\right.
\]

\[
  \mu_{|A|} = \mu_{|A^{*}|}, \qquad\text{i.e.}\qquad \varphi_{|A|} = \varphi_{|A^{*}|}
\]

\[
  T\bigl((AA^{*})^{p}\bigr) = T\bigl((A^{*}A)^{p}\bigr),
\]

\[
  \text{(1)}\qquad \Delta(A) = \struck{\ill{}}\ \exp\int_0^{T(1)} \log\varphi_{|A|} \qquad \text{\uncertain{pour} } A \in \tilde{\underline{b}} \text{ \uncertain{d'abord}, \uncertain{puis} } A \in \tilde{\underline{b}}
\]

\[
  \left\{
  \begin{array}{l}
    \Delta(A) = \Delta(A^{*}) \\
    \Delta(AB) = \Delta(A)\,\Delta(B) \\
    \Delta(1) = 1
  \end{array}\right.
  \qquad A, B \in 1 + \underline{a}
\]

\[
  \text{(5)}\qquad \struck{\Delta(1 + A) = \Delta_E(1_E + A_E)} \qquad \struck{(E = \text{\uncertain{support}} \ill{}}
\]

\[
  \text{(5')}\qquad \Delta(1 + A) = \Delta^{E}(1_E + A) \qquad \text{si } \struck{A \in \mathcal{A}_E}\ A \in \mathcal{A}_E
\]

\[
  \left\{
  \begin{array}{l}
    \Delta(A) = \Delta(A^{*}) \\
    \Delta(AB) = \Delta(A)\,\Delta(B) \\
    \Delta(1) = 1
  \end{array}\right.
  \qquad \text{si } A, B \in 1 + \underline{a}
\]

\begin{align*}
  \struck{\Delta(A)^{2}} &\struck{= \Delta(AA^{*})} \\
  \struck{\Delta(A^{*})^{2}} &\struck{= \Delta(A^{*}A)} \\
  \struck{\Delta(AB)^{2}} &\struck{= \Delta(B^{*}A^{*}AB) = \Delta(B^{*}HB) \overset{?}{=} \Delta(\uncertain{B^{*}A})\,\Delta(B^{*}B)}
\end{align*}

\[
  U^{*}U = X \qquad UU^{*} = Y
\]

\[
  \Delta^{X}(U^{*}AX) = \Delta^{Y}(YAU^{*})
\]

\[
  \text{(5)}\qquad \Delta^{X,Y}(A) = \Delta^{X}(U^{*}AX) = \Delta^{Y}(YAU^{*}) \qquad (\sigma(AX) \leq Y)
\]

\[
  \text{(6)}\qquad \Delta^{X,Z}(BA) = \Delta^{X,Y}(A)\,\Delta^{Y,Z}(B)
\]

\[
  \Delta_A(t) = \sup_{\operatorname{Tr} E \leq t} \Delta^{E}(EAE) \qquad (A \in \underline{b})\ (t \leq \operatorname{Tr} 1)
\]

\[
  \Delta_A(t) \geq \sup_{\operatorname{Tr} E \leq t} \Delta_E(EAE)
\]

\[
  \Delta^{E}(EAE) = \Delta^{E}_{EAE}(T(E)) = \Delta_{EAE}(t)\struck{\ill{}} \leq \Delta_A(t)\struck{\ill{}}
\]

\[
  \Delta^{EF}(FAE) \leq \Delta_A(t) \qquad (t = T(E))
\]

\[
  \Delta_{AB}(t) \leq \Delta_A(t)\,\Delta_B(t) \qquad (A, B \in \underline{b},\ \struck{t \leq |M|}\ t \geq 0)
\]

\[
  \Delta_{AB}(t) = \sup_{T(E) \leq t} \Delta_E(EABE)
\]

\[
  \Delta^{E}(EABE) \leq \Delta_A(t)\,\Delta_B(t)\,.
\]

\[
  \Delta^{E}(EABE) = \Delta^{E}(A'B') = \Delta^{E,F}(B')\,\Delta^{F,E}(A')\,.
\]

\begin{align*}
  \Delta^{E,F}(\uncertain{FBE}) &\leq \Delta_B(t) \\
  \Delta^{F,E}(EAF) &= \Delta_A(t)
\end{align*}

\[
  \Delta^{EF}(FAE) \leq \Delta_A(t)
\]

\[
  \Delta^{EF}(FAE) = \Delta^{E}(U^{*}FAE) = \Delta^{E}(U^{*}AE) \leq \Delta_{U^{*}A}(t) \leq \Delta_A(t) \quad \text{\uncertain{car}}\ \|U^{*}\| \leq 1
\]

\[
  \text{(1)}\qquad \Delta_{\varphi_{|AB|}} \leq \Delta_{\varphi_{|A|}\varphi_{|B|}}
\]

\[
  \text{(2)}\qquad W(AB) \leq W(\varphi_{|A|}\varphi_{|B|})
\]

\[
  \text{(3)}\qquad \int_0^{t} \varphi_{|AB|}(s)\, ds \leq \int_0^{t} \varphi_{|A|}(s)\,\varphi_{|B|}(s)\, ds
\]

\[
  \text{(4)}\qquad \boxed{\ \int \varphi_{|AB|} \leq \int \varphi_{|A|}\,\varphi_{|B|}\ }
\]

\[
  \text{(5)}\qquad |\operatorname{Tr} AB| \leq \struck{\ill{}}\ \operatorname{Tr}|AB| = \int \varphi_{|AB|} \leq \int \varphi_{|A|}\,\varphi_{|B|}
\]

\[
  \operatorname{Tr}|AB|^{r} \leq \int \bigl(\varphi_{|A|}\varphi_{|B|}\bigr)^{r} \qquad\text{i.e.}\qquad \|AB\|_r \leq \|\varphi_{|A|}\varphi_{|B|}\|_r
\]

\[
  \text{(6)}\qquad \|AB\|_r \leq \|A\|_p\,\|B\|_q \qquad \Bigl(\frac{1}{r} = \frac{1}{p} + \frac{1}{q}\Bigr)
\]

\[
  \text{(7)}\qquad \Phi_{|A|}(t) = \sup_{\sigma(B) \leq t} |\operatorname{Tr} AB| = \sup_{\substack{\operatorname{Tr}(\sigma(B)) \leq t\\ \|B\| \leq 1}} |\operatorname{Tr} BA|
\]

\[
  \operatorname{Tr} E|A| = \Phi_{|A|}(t), \quad \text{\uncertain{d'où}}\quad |\operatorname{Tr} EU^{*}A| = \Phi_{|A|}(t), \quad \text{\uncertain{et}}\quad \operatorname{Tr}\bigl(\sigma(EU^{*})\bigr) \leq \operatorname{Tr} E \leq t,\quad \|EU^{*}\| \leq 1.
\]

\[
  |\operatorname{Tr} BA| \leq \int \varphi_E\,\varphi_{|A|} = \int_0^{t} \varphi_{|A|}(s)\, ds = \Phi_{|A|}(t)\,.
\]

\[
  \text{(8)}\qquad \Phi_{|A+B|} \leq \Phi_{|A|} + \Phi_{|B|}
\]

\[
  \varphi_A(t+0) = \inf_{\substack{E \in \mathcal{P}\\ \operatorname{Tr} E < t}}\ \sup_{x \in (1-E)\mathcal{H}} (Ax, x) \qquad (\text{\uncertain{trace} \uncertain{continue}})
\]

\[
  \sup_{x \in (1-E)\mathcal{H}} (Ax, x) \geq \varphi_A^{-}(t)
\]

\[
  \mathcal{X} \ominus (\mathcal{X} \wedge \mathcal{Y}') \sim \mathcal{Y} \ominus (\mathcal{Y} \wedge \mathcal{X}') \qquad \text{\uncertain{donc}}\quad \mathcal{Y} \wedge \mathcal{X}' = 0
\]

\[
  \varphi_{|A|}(t+0) = \inf_{\substack{E \in \mathcal{P}\\ \operatorname{Tr} E < t}}\ \sup_{x \in (1-E)\mathcal{H}} \|Ax\| \qquad (\text{\uncertain{trace} \uncertain{continue}})
\]

\[
  \varphi_{|PA|}\struck{(t)} \leq \|P\|\,\varphi_{|A|}\struck{(t)} \qquad (\text{\uncertain{trace} \uncertain{quelconque} \ill{}}), \qquad
  \varphi_{|AQ|} \leq \|Q\|\,\varphi_{|A|}
\]

\[
  \mu_{|AA^{*}|}(x^{p}) = \mu_{|A^{*}A|}(x^{p}) \qquad \text{\uncertain{pour} \uncertain{tout} \ill{} \uncertain{entier} } p \geq 1,\ \text{\uncertain{soit}}
\]

\[
  \operatorname{Tr}\,\underbrace{AA^{*} \cdots AA^{*}}_{p} = \operatorname{Tr}\,\underbrace{A^{*}A \cdots A^{*}A}_{p}
\]

\[
  \Psi_{|A|}(t) = \Delta_t(A)
\]

\[
  \left\{
  \begin{array}{l}
    \Delta_t(A) = \displaystyle\sup_{\operatorname{Tr} E < t} \struck{\det}\ \Delta_E\, EAE \\[2ex]
    \Delta_t(1 + |A|) = \displaystyle\sup_{\substack{\operatorname{Tr}|B| < t\\ \|B\| \leq 1}} \struck{\det}\ \Delta_E\bigl(1_E + BA\uncertain{B}\bigr)
  \end{array}\right.
\]

\begin{align*}
  \Delta_E\, EAE &\leq \Delta_t(A) \qquad \text{\uncertain{si} } \operatorname{Tr} E \leq t \\
  \Delta_E(1_E + BAB) &\leq \Delta_t(1 + |A|)
\end{align*}

\begin{align*}
  \Delta_E A' &\leq \Delta_t A' \\
  \Delta_E(1 + B') &\leq \Delta_t(1 + |B'|)
\end{align*}

\[
  \varphi_{E(1+A)E} \leq \varphi_{1+A}\ )
\]

\[
  \Bigl(\Delta B = \exp\int \log\varphi_B(t) = \exp\int_0^{t} \log\varphi_B(s) = \Delta_t(B)\Bigr)
\]

\[
  \Delta_{\struck{t}}(1 + B) \leq \Delta(1 + |B|)
\]

\[
  \text{(*)}\qquad \|u^{*}u\| = \|u\|^{2}
\]

\[
  \|\,|u|.x\,\| = \|ux\| \ill{}, \quad \text{\uncertain{d'où}}\quad ux = U|u|x,
\]

\[
  \text{(***)}\qquad u = U|u| \qquad |u| = Vu \qquad (\|U\| \leq 1,\ \|V\| \leq 1)
\]

\[
  \text{(****)}\qquad u = \sum \rho_i\, \bar{e}_i \otimes f_i, \qquad \text{\uncertain{où} } f_i = Ue_i,
\]

\[
  u^{*} = \sum \rho_i\, \bar{f}_i \otimes e_i, \quad \text{\uncertain{donc}}\ \ill{}
\]

\[
  \rho_n(u) = \lambda_n(|u|) \qquad \text{\uncertain{pour} } u \in L_0(E, F).
\]

\[
  |\lambda_1(u)| \leq \rho_1(u) = \|u\| \qquad\qquad \rho_n(u) = \rho_n(u^{*})
\]

\[
  \rho_n(h) = \operatorname{Inf}_{E_{n-1}}\ \struck{\sup_{x \in F}\|hx\|}\ \sup_{\substack{x \in E_{n-1}^{\perp}\\ \|x\| \leq 1}} (hx, x)
\]

\[
  \rho_n(u)^{2} = \rho_n(u^{*}u) = \operatorname{Inf}_{F_n}\ \sup_{\substack{x \in F_n\\ \|x\| \leq 1}} \|ux\|^{2}, \quad \text{\uncertain{d'où} \uncertain{aussi}}
\]

\[
  = \operatorname{Inf}_{E_n}\ \sup_{\substack{x \in E_{n-1}\\ \|x\| \leq 1}} \|hx\|^{2} \qquad (h \geq 0) \struck{\ill{}}
\]

\[
  \rho_n(u) = \operatorname{Inf}_{F_n}\ \sup_{\substack{x \in E_{n-1}\\ \|x\| \leq 1}} \|ux\|
\]

\[
  \left\{
  \begin{array}{l}
    \rho_n(uv) \leq \|u\|\,\rho_n(v) \\
    \rho_n(uv) \leq \|v\|\,\rho_n(u)
  \end{array}\right.
\]

\[
  \left\{
  \begin{array}{l}
    \rho_{m+n-1}(AB) \leq \rho_m(A)\,\rho_n(B) \\
    \rho_{m+n-1}(A+B) \leq \rho_m(A) + \rho_n(B)
  \end{array}\right.
\]

\[
  \|u\|_1 = \sum \rho_i(u)
\]

\[
  |\operatorname{Tr} u| \leq \sum \rho_i(u)\,.
\]

\[
  \sum_{1}^{n} \rho_i(u) = \sup_{\substack{v_n \in L(F, E)\\ \|v_n\| \leq 1\\ \operatorname{rang} v_n \leq n}} |\operatorname{Tr} v_n u|
\]

\[
  \operatorname{Tr} p_n v_n = \operatorname{Tr} p_n v_n = \operatorname{Tr} p_n v_n p_n,
\]

\[
  \sum_{1}^{n} \rho_i(u + v) \leq \sum_{1}^{n} \bigl(\rho_i(u) + \rho_i(v)\bigr)
\]

\[
  \left\{
  \begin{array}{l}
    \|u\|_p = \Bigl(\displaystyle\sum \rho_i(u)^{p}\Bigr)^{1/p} \\[2ex]
    S_p(u) = \displaystyle\sum \bigl(\rho_i(u)\bigr)^{p}
  \end{array}\right.
\]

\[
  (u, v) = \operatorname{Tr} vu
\]

\[
  \text{(1)}\qquad \bar{\bar{E}} = E
\]

\[
  \text{(2)}\qquad \overline{E \otimes F} = \bar{E} \otimes \bar{F}
\]

\[
  \text{(3)}\qquad E' = \bar{E} \qquad (E \text{ espace de Hilbert})
\]

\[
  \text{(5)}\qquad (x_1 \otimes y_1,\; x_2 \otimes y_2) = (x_1, x_2)\,(y_1, y_2)
\]

\[
  \text{(6)}\qquad (e_i \otimes f_j,\; e_{i'} \otimes f_{j'}) = \delta_{(i,j),(i',j')} \qquad (\text{indice de Kronecker}),
\]

\[
  \text{(7)}\qquad \|x \otimes y\|_2 = \|x\|\,\|y\|
\]

\[
  \text{(8)}\qquad \overline{E \otimes^{(2)} F} = \bar{E} \otimes^{(2)} \bar{F}
\]

\[
  \text{(9)}\qquad \|f\|_2 = \Bigl(\int \struck{\ill{}}\, \|f(t)\|^{2}\, d\mu(t)\Bigr)^{1/2}
\]

\[
  \text{(10)}\qquad (f, g) = \int \langle f(t), g(t)\rangle\, d\mu(t)
\]

\[
  \text{(11)}\qquad L^{2}(\mu) \otimes^{(2)} E = L^{2}_{E}(\mu)
\]

\[
  \text{(12)}\qquad l^{2}(I) \otimes^{(2)} E = l^{2}_{E}(I)
\]

\[
  \text{(13)}\qquad L^{2}(\mu) \otimes^{(2)} L^{2}(\nu) = L^{2}(\mu \otimes \nu)
\]

\[
  \text{(14)}\qquad (\bar{a} \otimes b).x = (x, a).b
\]

\[
  \text{(15)}\qquad (ux, y) = (u,\; \bar{x} \otimes y)
\]

\[
  |(ux, y)| \leq \|u\|_2\, \|\bar{x} \otimes y\|_2 = \|u\|_2\, \|\bar{x}\|\, \|y\| = \|u\|_2\, \|x\|\, \|y\|,
\]

\[
  \text{(16)}\qquad \|u\|_{\infty} \leq \|u\|_2
\]

\[
  \text{(17)}\qquad \|Au\|_2 \leq \|A\|\,\|u\|_2 \qquad (\text{resp. } \|uA\|_2 \leq \|A\|\,\|u\|_2)
\]

\[
  \text{(18)}\qquad \struck{(u, v) = (v^{*}, u^{*})}
\]

\[
  \text{(19)}\qquad \|u\|_2 = \|u^{*}\|_2 = \Bigl(\sum \rho_i^{2}\Bigr)^{1/2}
\]

\[
  \text{(20)}\qquad \|u\|_1 = \|h\|_1 = \sum \rho_i
\]

\[
  \sum_{1}^{n} \rho_i = \operatorname{Tr} p_n h \leq \|p_n h\|_1 \leq \|h\|_1,
\]

\[
  u = \sum \rho_i\, \bar{a}_i \otimes b_i,
\]

\[
  u^{*} = \sum \rho_i\, \bar{b}_i \otimes a_i, \quad \text{et}\quad uu^{*} = \sum_{i,j} \rho_i \rho_j\, (a_i, a_j)\, \bar{b}_j \otimes b_i = \sum \rho_i^{2}\, \bar{b}_i \otimes b_i,
\]

\[
  \text{d'où}\quad \sqrt{uu^{*}} = \sum \rho_i\, \bar{b}_i \otimes b_i,
\]

\[
  \text{(21)}\qquad u = \sum \rho_i\, \bar{a}_i \otimes b_i
\]

\[
  \text{(22)}\qquad \|u\|_1 = \operatorname{Tr} \sqrt{u^{*}u} = \operatorname{Tr} \sqrt{uu^{*}}
\]

\[
  \text{(23)}\qquad \|u\|_1 = \sup_{\|A\| \leq 1} |\operatorname{Tr} Au| \qquad (A \in L(F,E))
\]

\[
  \text{(24)}\qquad \|vu\|_1 \leq \|v\|_2\, \|u\|_2
\]

\[
  \text{(25)}\qquad (u, v) = \operatorname{Tr} uv^{*} = \operatorname{Tr} v^{*}u
\]

\[
  u = \bar{a} \otimes b, \quad v = \bar{a}' \otimes b', \quad \text{d'où}\quad v^{*} = \bar{b}' \otimes a',
\]

\[
  v^{*}u = (b, b')\, \bar{a} \otimes a', \qquad \struck{uv^{*} = (a', a)\, \bar{b} \otimes \ill{}} \qquad \text{d'où}
\]

\[
  (u, v) = (\bar{a}, \bar{a}')\,(b, b') = \struck{(a^{*}, a')}\,(b, b'), \qquad \operatorname{Tr} v^{*}u = (a', a)\,(b, b').
\]

\[
  \text{(26)}\qquad \|u\|_2 = \sup_{\|v\|_2 \leq 1} |\operatorname{Tr} vu| \qquad (v \in L^{(2)}(F,E))
\]

\[
  \|u\|_2 = \sup_{\|w\|_2 \leq 1} |\operatorname{Tr} w^{*}u| \qquad (w \in L^{(2)}(E,F))
\]

\[
  \text{(1)}\qquad (a_1 \otimes \cdots \otimes a_n,\; b_1 \otimes \cdots \otimes b_n) = (a_1, b_1) \cdots (a_n, b_n)
\]

\[
  \text{(2)}\qquad \|u_1 \otimes \cdots \otimes u_n\| \leq \|u_1\| \cdots \|u_n\|
\]

\[
  \text{(3)}\qquad a_n = \frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \varepsilon_\sigma\, \sigma
\]

\[
  (a_{\sigma 1} \otimes \cdots \otimes a_{\sigma n},\; b_1 \otimes \cdots \otimes b_n) = (a_1 \otimes \cdots \otimes a_n,\; b_{\sigma^{-1} 1} \otimes \cdots \otimes b_{\sigma^{-1} n})
\]

\[
  \struck{(a_1 \wedge \cdots \wedge a_n,\; b_1 \wedge \cdots \wedge b_n) = \Bigl(\frac{1}{n!}\Bigr)^{2} \sum_{\sigma \in \mathfrak{S}_n} \varepsilon_\sigma\, \sigma\, a_1 \ldots}
\]

\[
  (a_1 \wedge \cdots \wedge a_n,\; b_1 \wedge \cdots \wedge b_n) = n!\,(a_n A,\; a_n B) = n!\,(A,\; a_n^{2} B) = n!\,(A,\; a_n B)
\]

\[
  = \sum_{\sigma \in \mathfrak{S}_n} \varepsilon_\sigma\, (a_1, b_{\sigma 1}) \cdots (a_n, b_{\sigma n}) = \det\bigl((a_i, b_j)\bigr), \qquad \text{donc}
\]

\[
  \text{(4)}\qquad (a_1 \wedge \cdots \wedge a_n,\; b_1 \wedge \cdots \wedge b_n) = \det\bigl((a_i, b_j)\bigr)_{i,j \leq n}
\]

\[
  \text{(5)}\qquad \|a_1 \wedge \cdots \wedge a_n\| = \bigl(\det((a_i,a_j))\bigr)^{1/2} \leq \|a_1\| \cdots \|a_n\| \qquad (a_i \in E)
\]

\[
  \det\bigl((a_i,a_j)\bigr) = \det A A^{*} = \det A \cdot \det A^{*} = \det A \cdot \overline{\det A} = |\det A|^{2},
\]

\[
  \text{(6)}\qquad |\det A| \leq \|a_1\| \cdots \|a_n\| \qquad \text{(inégalité de Hadamard)},
\]

\[
  \bigl|\det\bigl((a_i, b_j)\bigr)\bigr| \leq \prod_i \|a_i\|\, \|b_i\| .
\]

\[
  \text{(5)}\qquad e_J = e_{i_1} \wedge \cdots \wedge e_{i_n} .
\]

\[
  \text{(8)}\qquad u_1 \wedge \cdots \wedge u_n = n!\; \varphi_n \cdot (u_1 \otimes \cdots \otimes u_n) \cdot a_n
\]

\[
  \|u_1 \wedge \cdots \wedge u_n\| = \Bigl\|\sqrt{n!}\; \varphi_n \cdot (u_1 \otimes \cdots \otimes u_n) \cdot \tfrac{1}{\sqrt{n!}}\, a_n\Bigr\| \leq \|u_1 \otimes \cdots \otimes u_n\|,
\]

\[
  \text{(9)}\qquad \|u_1 \wedge \cdots \wedge u_n\| \leq \|u_1\| \cdots \|u_n\| .
\]

\[
  \text{(10)}\qquad (\Lambda v)(\Lambda u) = \Lambda(vu)
\]

\[
  \text{(11)}\qquad \struck{(\Lambda u)^{*} = \ldots}\qquad (u_1 \wedge \cdots \wedge u_n)^{*} = u_1^{*} \wedge \cdots \wedge u_n^{*}
\]

\begin{gather}
  \bigl((\Lambda u)\, a_1 \wedge \cdots \wedge a_n,\; b_1 \wedge \cdots \wedge b_n\bigr) = \det\bigl((u a_i, b_j)\bigr) = \det\bigl((a_i, u^{*} b_j)\bigr) \\
  = \bigl(a_1 \wedge \cdots \wedge a_n,\; (\Lambda u^{*})\, b_1 \wedge \cdots \wedge b_n\bigr),
\end{gather}

\[
  \text{(12)}\qquad (\Lambda h)^{\alpha} = \Lambda(h^{\alpha}) \qquad (h \in L(E),\; h \geq 0)
\]

\[
  \sum_{0 \leq p \leq n} \Lambda^{p} E_0 \otimes \Lambda^{n-p} E_1
\]

\[
  \Lambda E \cong \sum_{p+q=n} \Lambda^{p} E_0 \otimes \Lambda^{q} E_1
\]

\[
  \text{(23)}\qquad |\lambda_1(u) \cdots \lambda_n(u)| \leq \rho_1(u) \cdots \rho_n(u)
\]

\[
  \text{(21)}\qquad \lambda_1(u) \cdots \lambda_n(u) = \lambda_1(\Lambda^{n} u)
\]

\[
  \text{(22)}\qquad \rho_1(u) \cdots \rho_n(u) = \|\Lambda^{n} u\|
\]

\[
  \text{(24)}\qquad \rho_1(uv) \cdots \rho_n(uv) \leq \bigl(\rho_1(u)\, \rho_1(v)\bigr) \cdots \bigl(\rho_n(u)\, \rho_n(v)\bigr)
\]

\[
  \bigl(\rho_1(uv) \cdots \rho_n(uv)\bigr)^{2} = \lambda_1\bigl((uv)(uv)^{*}\bigr) \cdots \lambda_n\bigl((uv)(uv)^{*}\bigr) = \lambda_1\bigl(\Lambda((uv)(uv)^{*})\bigr) \leq \ill{}
\]

\[
  \text{(1)}\qquad \|u_1 \wedge \cdots \wedge u_n\|_1 \leq \frac{1}{n!}\, \|u_1\|_1 \cdots \|u_n\|_1
\]

\[
  \text{(2)}\qquad \alpha_n(u_1, \ldots, u_n) = \operatorname{Tr}(u_1 \wedge \cdots \wedge u_n)
\]

\[
  \text{(3)}\qquad \alpha_n(u) = \operatorname{Tr} \Lambda^{n}(u)
\]

\[
  \text{(4)}\qquad |\alpha_n(u_1, \ldots, u_n)| \leq \frac{1}{n!}\, \|u_1\|_1 \cdots \|u_n\|_1 \qquad \text{---}\qquad |\alpha_n(u)| \leq \frac{1}{n!}\, \|u\|_1^{n}
\]

\[
  \text{(5)}\qquad |\alpha_n(u)| \leq \struck{\ldots}\; \alpha_n(|u|) = \|\Lambda u\|_1 \leq \frac{1}{n!}\, \|u\|_1^{n}
\]

\[
  \text{(6)}\qquad |\det(1+u)| \leq \det(1+|u|)
\]

\[
  \text{(7)}\qquad \det(1+zu) = \prod_i \bigl(1 + z\,\lambda_i(u)\bigr)
\]

\[
  \text{(8)}\qquad \alpha_n(u) = \sum_{i_1 < \cdots < i_n} \lambda_{i_1}(u) \cdots \lambda_{i_n}(u) \qquad (\uncertain{\text{première}} \ill{} \Longrightarrow
\]

\[
  \text{(9)}\qquad \operatorname{Tr} u = \sum_i \lambda_i(u)
\]

\[
  \text{(10)}\qquad \alpha_n(|u|) = \|\Lambda u\|_1 = \sum_{i_1 < \cdots < i_n} \rho_{i_1}(u) \cdots \rho_{i_n}(u)
\]

\[
  \text{(11)}\qquad \|(\Lambda^{p} u) \wedge (\Lambda^{q} v)\|_1 \leq \frac{p!\, q!}{(p+q)!}\, \|\Lambda^{p} u\|_1\, \|\Lambda^{q} v\|_1 = \frac{p!\, q!}{(p+q)!}\, \alpha_p(|u|)\, \alpha_q(|v|)
\]

\begin{gather}
  (\Lambda^{p} u) \wedge (\Lambda^{q} v) = \frac{1}{(p+q)!} \sum_{\substack{i_1 < \cdots < i_p \\ j_1 < \cdots < j_q}} \rho_{i_1} \cdots \rho_{i_p}\, \sigma_{j_1} \cdots \sigma_{j_q}\; \times \\
  \overline{b_{i_1} \wedge \cdots \wedge b_{i_p} \wedge d_{j_1} \wedge \cdots \wedge d_{j_q}} \otimes \bigl(a_{i_1} \wedge \cdots \wedge a_{i_p} \wedge c_{j_1} \wedge \cdots \wedge c_{j_q}\bigr)
\end{gather}

\begin{gather}
  \|(\Lambda^{p} u) \wedge (\Lambda^{q} v)\|_1 \leq \frac{1}{(p+q)!}\, p!\, q! \sum \rho_{i_1} \cdots \rho_{i_p}\, \sigma_{j_1} \cdots \sigma_{j_q} \\
  = \frac{p!\, q!}{(p+q)!} \Bigl(\sum_{i_1 < \cdots < i_p} \rho_{i_1} \cdots \rho_{i_p}\Bigr) \Bigl(\sum_{j_1 < \cdots < j_q} \sigma_{j_1} \cdots \sigma_{j_q}\Bigr) = \frac{p!\, q!}{(p+q)!}\, \|\Lambda^{p} u\|_1\, \|\Lambda^{q} v\|_1
\end{gather}

\[
  \text{(12)}\qquad \struck{\det}\; \det(1 + r|u+v|) \leq \det(1 + r|u|) \bigl(\det(1 + r|v|)\bigr)
\]

\[
  \text{(13)}\qquad \prod_i \bigl(1 + r\rho_i(u+v)\bigr) \leq \Bigl(\prod_i \bigl(1 + r\rho_i(u)\bigr)\Bigr) \Bigl(\prod_i \bigl(1 + r\rho_i(v)\bigr)\Bigr)
\]

\[
  \text{(14)}\qquad \alpha_n(|u+v|) \leq \sum_{0 \leq p \leq n} \alpha_p(|u|)\, \alpha_{n-p}(|v|)
\]

\[
  \Lambda^{n}(u+v) = \sum_{p=0}^{n} \frac{n!}{p!\,(n-p)!}\, (\Lambda^{p} u) \wedge (\Lambda^{n-p} v), \qquad \text{d'où}
\]

\[
  \alpha_n(|u+v|)\; \|\Lambda^{n}(u+v)\|_1 \leq \sum_{p=0}^{n} \frac{n!}{p!\,(n-p)!}\, \|(\Lambda^{p} u) \wedge (\Lambda^{n-p} v)\|_1 \leq \sum \struck{\frac{n!}{p!\,(n-p)!}}\; \alpha_p(|u|)\, \alpha_{n-p}(|v|).
\]

\[
  \text{(15)}\qquad \prod_{1}^{n} \bigl(1 + r\rho_i(u+v)\bigr) \leq \Bigl(\prod_{1}^{n} \bigl(1 + r\rho_i(u)\bigr)\Bigr) \Bigl(\prod_{1}^{n} \bigl(1 + r\rho_i(v)\bigr)\Bigr)
\]

\[
  \text{(16)}\qquad \sigma_n^{m}(u+v) \leq \sum_{p=0}^{m} \sigma_n^{p}(u)\, \sigma_n^{m-p}(v)
\]

\[
  \text{(17)}\qquad \sigma_n^{m}(w) = \sum_{1 \leq i_1 < \cdots < i_m \leq n} \rho_{i_1}(w) \cdots \rho_{i_m}(w)
\]

\[
  \text{(19)}\; \struck{\text{(18)}}\qquad \sigma_n^{m}(u) = \sup_{\substack{v_n \in L(F,E) \\ \|v_n\| \leq 1,\; \operatorname{rang} v_n \leq n}} |\alpha_m(v_n u)| = \sup_{v_n} \{\alpha_m(|v_n u|)\}
\]

\[
  \struck{|\alpha_m(v_n u)| = |\operatorname{Tr} \Lambda^{m} v_n u| \leq \ill{}} \qquad \struck{\leq \|\Lambda^{m} \ill{}\| \cdot \|\Lambda^{m} u\|_1 \leq \ill{}}
\]

\[
  |\alpha_m(u)| \leq \{\alpha_m(|u|)\} \leq \sigma_n^{m}(u).
\]

\[
  |\alpha_m(u)| = |\alpha_m(w)| \leq \struck{\ill{}} \sum_{1 \leq i_1 < \cdots < i_m \leq n} \rho_{i_1}(w) \cdots \rho_{i_m}(w)
\]

\[
  |\alpha_m(w(u+v))| \leq \alpha_m(|wu + wv|) \leq \sum_{p=0}^{m} \alpha_p(|wu|)\, \alpha_{m-p}(|wv|)
\]

\[
  \text{(18)}\qquad \struck{\det}\; \prod_{1}^{n} \bigl(1 + \rho_i(u)\bigr) = \sup_{\substack{v_n \in L(F,E) \\ \|v_n\| \leq 1,\; \operatorname{rang} v_n \leq n}} \det(1 + v_n u)
\]

\[
  \sum_{1}^{n} \log\bigl(1 + r\rho_i(u+v)\bigr) \leq \sum_{1}^{n} \log\bigl(1 + r\rho_i(u)\bigr) + \sum_{1}^{n} \log\bigl(1 + r\rho_i(v)\bigr)
\]

\[
  S_\varphi^{n}(u+v) \leq S_\varphi^{n}(u) + S_\varphi^{n}(v), \qquad S_\varphi^{n}(w) = \sum_{1}^{n} f_\varphi\bigl(\rho_i(w)\bigr)
\]

\[
  f_\varphi(\rho) = \int_{0}^{+\infty} \frac{\varphi(r)}{\frac{1}{\rho} + r}\, dr = \int_{0}^{+\infty} \varphi(s/\rho)\, \frac{ds}{1+s}
\]

\[
  -\int \varphi'(r)\, A(r)\, dr = \bigl[-\varphi(r)\, A(r)\bigr]_{0}^{+\infty} + \int \varphi(r)\, A'(r)\, dr
\]

\[
  f_p(\rho) = \rho^{p} \int \frac{s^{-p}}{1+s}\, ds = c^{te} \cdot \rho^{p}, \qquad \text{d'où, en posant}
\]

\[
  S_p^{(n)}(u) = \sum_{1}^{n} \rho_i(u)^{p}
\]

\[
  S_p^{(n)}(u+v) \leq S_p^{(n)}(u) + S_p^{(n)}(v) \qquad \text{en particulier}
\]

\[
  S_p(u+v) \leq S_p(u) + S_p(v)
\]

\[
  f^{(n)}(u+v) \leq f^{(n)}(u) + f^{(n)}(v) \qquad \text{quel que soit } n.
\]

\[
  \text{(13)}\qquad \|u_1 \wedge \cdots \wedge u_n\|_1 \leq \frac{1}{n!}\, \|u_1\|_1 \cdots \|u_n\|_1
\]

\[
  \struck{\ill{}}\quad u_1 \wedge \cdots \wedge u_n = \frac{1}{n!}\, (\bar{b}_1 \wedge \cdots \wedge \bar{b}_n) \otimes (a_1 \wedge \cdots \wedge a_n), \qquad \text{d'où}
\]

\[
  \|u_1 \wedge \cdots \wedge u_n\|_1 \leq \frac{1}{n!}\, \|b_1 \wedge \cdots \wedge b_n\|\, \|a_1 \wedge \cdots \wedge a_n\| \leq \frac{1}{n!} \prod \|a_i\|\, \|b_i\|
\]

\[
  \text{(14)}\qquad \alpha_n(u_1, \ldots, u_n) = \operatorname{Tr}(u_1 \wedge \cdots \wedge u_n), \qquad \alpha_n(u) = \operatorname{Tr}(\Lambda^{n} u) \quad (\uncertain{\text{en particulier}})
\]

\[
  \text{(15)}\qquad |\alpha_n(u_1, \ldots, u_n)| \leq \frac{1}{n!}\, \|u_1\|_1 \cdots \|u_n\|_1
\]

\[
  \text{(16)}\qquad \|\Lambda^{n} u\|_1 = \alpha_n(|u|) = \alpha_n\bigl(\sqrt{u^{*}u}\bigr)
\]

\[
  \text{(17)}\qquad \|\Lambda^{n} u\|_2 = \sqrt{\alpha_n(u^{*}u)}
\]

\[
  \text{(18)}\qquad |\alpha_n(u)| \leq \alpha_n(|u|)
\]

\[
  \text{(19)}\qquad |\det(1+zu)| \leq \det(1 + |z|\,|u|) \qquad \struck{(\ill{})}
\]

\[
  \text{(20)}\qquad \det(1+u) = \prod_i (1 + \lambda_i)
\]

\[
  \text{(21)}\qquad \det(1+zu) = \prod_i (1 + z\lambda_i)
\]

\[
  \text{(22)}\qquad \alpha_n(u) = \sum_{i_1 < \cdots < i_n} \lambda_{i_1} \cdots \lambda_{i_n}
\]

\[
  \text{(23)}\qquad \operatorname{Tr} u = \sum \lambda_i(u)
\]

\[
  |z_1 \cdots z_n| = |\operatorname{Tr} pu| \leq \|pu\|_1 \leq \|p\|\, \|u\|_1 = \|u\|_1 .
\]

\[
  \alpha_n(u) = \sum_{i_1 \ldots i_n} \struck{\alpha_n(z_i e \ldots)}\; \lambda_{i_1} \cdots \lambda_{i_n}\; \alpha_n\bigl(\bar{e}_{i_1} \otimes e_{i_1}, \ldots, \bar{e}_{i_n} \otimes e_{i_n}\bigr)
\]

\[
  (x,y) \in G_f \iff x \in U \text{ et } y \geq f(x)
\]

\[
  \text{4)}\qquad f(x) = \sup_a L_a(x) \qquad (\ill{})
\]

\[
  f(x_1) \leq f(a) \leq f(a_1)
\]

\[
  \text{(1)}\qquad \alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n
\]

\[
  \text{(2)}\qquad \alpha_1 + \cdots + \alpha_m \leq \beta_1 + \cdots + \beta_m \qquad \text{pour } m = 1, \ldots, n,
\]

\[
  \text{(3)}\qquad \varphi(\alpha_1, \ldots, \alpha_n) \leq \varphi(\beta_1, \ldots, \beta_n) .
\]

\[
  \text{(4)}\qquad \rho_1 \geq \rho_2 \geq \cdots \geq \rho_n
\]

\[
  \text{(5)}\qquad \rho_1 \cdots \rho_m \leq \sigma_1 \cdots \sigma_m \qquad \text{pour tout } m = 1, \ldots, n .
\]

\[
  \text{(6)}\qquad f(\rho_1, \ldots, \rho_n) \leq f(\sigma_1, \ldots, \sigma_n) .
\]

\[
  \text{(7)}\qquad \struck{\varphi(t)}\; \varphi = \operatorname{Sup}_k \operatorname{Sup}_{\sigma \in \mathfrak{S}_n} \sigma . L_k \struck{(t_1, \ldots, t_n)}
\]

\[
  \text{(8)}\qquad \operatorname{Sup}_{\sigma \in \mathfrak{S}_n} (\sigma . L)(\alpha_1, \ldots, \alpha_n) \leq \operatorname{Sup}_{\sigma \in \mathfrak{S}_n} (\sigma L)(\beta_{\sigma 1}, \ldots, \beta_{\sigma n})
\]

\[
  \sum a_i \alpha_i \leq \sum a_i \alpha_{\sigma i} \qquad \struck{\ill{}} \ill{}
\]

\[
  \sum a_{\sigma i}\, \alpha_i \quad (\ill{} \uncertain{maximum} \ill{}), \qquad \text{pour } \sigma \in \mathfrak{S}_n \text{ variable},
\]

\[
  \sum a_i \alpha_i \leq \sum a_i \beta_i ,
\]

\begin{align}
  \text{(9)}\qquad \sum a_i \alpha_i &= (a_1 - a_2)\,\alpha_1 + (a_2 - a_3)(\alpha_1 + \alpha_2) + \cdots \notag \\
  &\qquad + (a_{n-1} - a_n)(\alpha_1 + \cdots + \alpha_{n-1}) + a_n(\alpha_1 + \cdots + \alpha_n) \notag
\end{align}

\[
  \text{(10)}\qquad f\Bigl(\frac{1}{\rho_1}, \ldots, \frac{1}{\rho_n}\Bigr) \leq f\Bigl(\frac{1}{\sigma_1}, \ldots, \frac{1}{\sigma_n}\Bigr)
\]

\[
  \rho_M \cdots \rho_n = \frac{\sigma_1 \cdots \sigma_{M-1}}{\rho_1 \cdots \rho_{M-1}}\; \sigma_M \cdots \sigma_n ,
  \qquad \text{d'où}\quad \rho_M \cdots \rho_n \;\uncertain{\geq}\; \sigma_M \cdots \sigma_n
\]

\[
  \varphi(\alpha_1, \ldots, \alpha_k) \leq \varphi(\beta_1, \ldots, \beta_k) .
\]

\[
  \text{(\uncertain{12})}\qquad N(x_1, \ldots, x_n) = N(|x_1|, \ldots, |x_n|) \qquad \struck{\ill{} \uncertain{fonction} \uncertain{croissante} \uncertain{de} \uncertain{chaque} \ill{}}
\]

\[
  \text{(15)}\qquad \sum_1^n \rho_i^p \leq \sum_1^n \sigma_i^p \qquad (p \text{ réel})
\]

\[
  \text{(16)}\qquad \sum_1^k \rho_i^p \leq \sum_1^k \sigma_i^p \qquad (p \geq 0,\; \struck{\ill{}} k = 1, \ldots, n) .
\]

\[
  \text{(17)}\qquad f(|\lambda_1|, \ldots, |\lambda_n|) \leq f(\rho_1, \ldots, \rho_n)
\]

\[
  \text{(13)}\qquad \sum_1^n f(\rho_i) \leq \sum_1^n f(\sigma_i)
\]

\[
  \text{(14)}\qquad \sum_1^k f(\rho_i) \leq \sum_1^k f(\sigma_i) \qquad (k = 1, \ldots, n)
\]

\[
  \text{(18)}\qquad \struck{f(\rho)}\; \sum_1^k f(|\lambda_i|) \leq \sum_1^k f(\rho_i) \qquad (k = 1, \ldots) \; \struck{(k \leq \dim E)}
\]

\[
  \text{(19)}\qquad \sum_{i=1}^\infty f(|\lambda_i|) \leq \sum_{i=0}^\infty f(\rho_i)
\]

\[
  \text{(20)}\qquad \sum_{i=1}^k |\lambda_i|^p \leq \sum_{i=1}^k \rho_i^p = \struck{\|u\|_p} \qquad \struck{(k \leq \dim E)}
\]

\[
  \text{(21)}\qquad \sum_i |\lambda_i|^p \leq \sum_i \rho_i^p = \|u\|_p
\]

\[
  \text{(21)}\qquad \sum |\lambda_i| \leq \|u\|_1
\]

\[
  \text{(22)}\qquad \Bigl(\sum |\lambda_i|^2\Bigr)^{1/2} \leq \|u\|_2
\]

\[
  \text{(23)}\qquad f(|\lambda_1|, \ldots, |\lambda_n|) \leq f(\rho_1, \ldots, \rho_n) .
\]

\[
  |\lambda_1 \cdots \lambda_k| \leq \rho_1 \cdots \rho_k \qquad (k \leq n),
\]

\[
  |\lambda_1 \cdots \lambda_n|^2 = |\det u|^2 = \ill{} = \det u^{*} u = \rho_1^2 \cdots \rho_n^2 \quad (\text{\uncertain{de} } u^{*} u)
\]

\[
  \text{(27)}\qquad N(|\lambda_1|, \ldots, |\lambda_k|) \leq N(\rho_1, \ldots, \rho_k)
\]

\[
  \text{(28)}\qquad N\bigl((|\lambda_i|)\bigr) \leq N\bigl((\rho_i)\bigr)
\]

\[
  N(\xi) = N\bigl((\xi_i)\bigr) = \lim N(x_n), \qquad \ill{} \text{ limite croissante } \geq 0 .
\]

\[
  \text{(29)}\qquad f\bigl(\rho_1(vu), \ldots, \rho_n(vu)\bigr) \leq f\bigl(\rho_1(v)\rho_1(u), \ldots, \rho_n(v)\rho_n(u)\bigr)
\]

\[
  \text{(30)}\qquad \sum_{i=1}^n f\bigl(\rho_i(vu)\bigr) \leq \sum_{i=1}^n f\bigl(\rho_i(v)\rho_i(u)\bigr)
\]

\[
  \text{(31)}\qquad \sum_i f\bigl(\rho_i(vu)\bigr) \leq \sum_i f\bigl(\rho_i(v)\rho_i(u)\bigr)
\]

\[
  \text{(34)}\qquad \frac{1}{p} + \frac{1}{q} = \frac{1}{s} \qquad (p, q, s \geq 0)
\]

\[
  \Bigl(\sum_{i=1}^\infty \rho_i(vu)^s\Bigr)^{1/s} \leq \Bigl(\sum_{i=1}^\infty \bigl(\rho_i(u)\rho_i(v)\bigr)^s\Bigr)^{1/s}
\]

\[
  \Bigl(\sum \rho_i(u)^p\Bigr)^{1/p} \Bigl(\sum \rho_i(v)^q\Bigr)^{1/q} .
\]

\[
  \text{(35)}\qquad \Bigl(\sum_{i=1}^n \rho_i(vu)^s\Bigr)^{1/s} \leq \Bigl(\sum_{i=1}^n \rho_i(u)^p\Bigr)^{1/p} \Bigl(\sum_{i=1}^n \rho_i(v)^q\Bigr)^{1/q}
\]

\[
  \text{(36)}\qquad \Bigl(\sum \rho_i(vu)^s\Bigr)^{1/s} \leq \Bigl(\sum \rho_i(u)^p\Bigr)^{1/p} \Bigl(\sum \rho_i(v)^q\Bigr)^{1/q}
\]

\[
  \struck{N_s(vu) \leq N_p(u)\, N_q(v)} \qquad \|vu\|_s \leq \|u\|_p\, \|v\|_q \qquad \Bigl(\frac{1}{s} = \frac{1}{p} + \frac{1}{q}\Bigr)
\]

\[
  \text{(38)}\qquad \|vu\|_1 \leq \|u\|_p\, \|v\|_{p'} \qquad \Bigl(\frac{1}{p} + \frac{1}{p'} = 1\Bigr)
\]

\[
  \text{(39)}\qquad f\bigl(\rho_1(vu), \ldots, \rho_n(vu)\bigr) \leq f\bigl(\rho_1(u)\rho_1(v), \ldots, \rho_n(u)\rho_n(v)\bigr)
\]

\[
  \text{(40)}\qquad \prod \rho_i(vu) = \struck{f(vu)}\; \prod \rho_i(u)\rho_i(v)
\]

\[
  \text{(41)}\qquad \varphi\bigl(\rho_1(u+v), \ldots, \rho_n(u+v)\bigr) \leq \varphi\bigl(\rho_1(u)+\rho_1(v), \ldots, \rho_n(u)+\rho_n(v)\bigr)
\]

\[
  \struck{\text{(42)}\qquad \sum_1^n \varphi\bigl(\rho_i(u+v)\bigr) \leq \sum_1^n \varphi\bigl(\rho_i(u) + \rho_i(v)\bigr)}
\]

\[
  \text{(43)}\qquad \|u\|_N = N\bigl(\rho_1(u), \ldots, \rho_n(u)\bigr)
\]

\[
  \text{(44)}\qquad \struck{\|\lambda u\|_N = |\lambda|\, \|u\|_N} \qquad \|u + v\|_N \leq \|u\|_N + \|v\|_N
\]

\[
  N\bigl(\rho_1(u+v), \ldots, \rho_n(u+v)\bigr) \leq N\bigl(\rho_1(u)+\rho_1(v), \ldots, \rho_n(u)+\rho_n(v)\bigr)
\]

\[
  \text{(45)}\qquad \Bigl(\sum_1^n \bigl(\rho_i(u+v)\bigr)^p\Bigr)^{1/p} \leq \Bigl(\sum_1^n \rho_i(u)^p\Bigr)^{1/p} + \Bigl(\sum_1^n \rho_i(v)^p\Bigr)^{1/p}
\]

\[
  \text{(46)}\qquad \|u + v\|_p \leq \|u\|_p + \|v\|_p
\]

\[
  \text{(1)}\qquad N(x_1, \ldots, x_n) = N(|x_1|, \ldots, |x_n|)
\]

\[
  \text{(2)}\qquad \|(x_i)\|_p = \Bigl(\sum |x_i|^p\Bigr)^{1/p} \qquad (1 \leq p < +\infty)
\]

\[
  \text{(3)}\qquad \|(x_i)\|_\infty = \operatorname{Sup}_i |x_i|
\]

\[
  \text{(4)}\qquad N(x) = \lim_n N(|x|_n)
\]

\[
  \text{(5)}\qquad N(\lambda x) = |\lambda|\, N(x), \qquad N(x+y) \leq N(x) + N(y),
\]

\[
  \text{(6)}\qquad \ell^1 \subset \ell^N \subset \ell^\infty
\]

\[
  \text{(7)}\qquad \struck{\sum}\; N\Bigl(\sum_k x^k\Bigr) \leq \sum_k N(x^k)
\]

\[
  N\Bigl(x - \sum_{k=1}^n x^k\Bigr) = \struck{\ill{}}\; N\Bigl(\struck{x} \sum_{n+1}^\infty x^k\Bigr) \leq \sum_{n+1}^\infty N(x^k),
\]

\[
  \text{(8)}\qquad \ell^1 \subset \ell_N \subset c_0
\]

\[
  \struck{\text{(9)}\qquad \ell_N = \ell^N \cap c_0}
\]

\[
  \text{(9)}\qquad N^{\circ}(y) = \operatorname{Sup}_{N(x) \leq 1} |\langle x, y \rangle|
\]

\[
  \text{(10)}\qquad \sum |x_i x_i'| \leq N(x)\, N'(x')
\]

\[
  \text{(11)}\qquad N(u) = N\bigl((\rho_i(u))\bigr) < +\infty
\]

\[
  \bigl(L_N(E,F)\bigr)' = L^{N'}(F,E) .
\]

\[
  \|vu\|_1 \leq N(u)\, N'(v)
\]

\[
  \text{(\uncertain{15})}\qquad N(xy) \leq N'(x)\, N''(y)
\]

\[
  \text{(16)}\qquad N(vu) \leq N'(u)\, N''(v)
\]

\[
  \text{(17)}\qquad \|uv\|_1 \leq N(u)\, N^{\circ}(v)
\]

\[
  N(u) = N(|u|)
\]

\[
  u \in L_N(E) \iff h, k \in L_N(E), \qquad u \in L^N(E) \iff h, k \in L^N(E) .
\]

\[
  N((\rho_i(u))) = \operatorname{Sup}_{v \in F' \otimes E,\ N^{\circ}(v) \leq 1} |\operatorname{Tr} vu| .
\]

\[
  \text{(18)}\qquad N(u) = N((\rho_i(u)))
\]

\[
  \text{(19)}\qquad N(u) = \operatorname{Sup}_{v \in F' \otimes E,\ N^{\circ}(v) \leq 1} |\operatorname{Tr} vu|
\]

\[
  \text{(20)}\qquad N(u) = N(u^{*})
\]

\[
  \text{(21)}\qquad N(uv) \leq \|u\|\,N(v), \qquad N(vu) \leq \|v\|\,N(u)
\]

\[
  \text{(22)}\qquad N(xy) \leq N'(x)\,N''(y)
\]

\[
  \text{(23)}\qquad N(vu) \leq N'(u)\,N''(v)
\]

\[
  \text{(24)}\qquad \|vu\|_1 \leq N(u)\,N^{\circ}(v)
\]

\[
  \text{(25)}\qquad |\operatorname{Tr} vu| \leq N(u)\,N^{\circ}(v) .
\]

\[
  \text{(26)}\qquad \langle u, v \rangle = \operatorname{Tr} vu
\]

\[
  \text{(27)}\qquad N((\lambda_i(u))) \leq N(u)
\]

\[
  \text{(1)}\qquad f(z) = e^{g(z)} \prod_n \Bigl(1 - \frac{z}{a_n}\Bigr)\, e^{\frac{z}{a_n} + \frac{z^2}{2a_n^2} + \cdots + \frac{z^{p_n}}{p_n a_n^{p_n}}}
\]

\[
  \text{(2)}\qquad f(z) = e^{g(z)} \prod_n \Bigl(1 - \frac{z}{a_n}\Bigr)\, e^{\frac{z}{a_n} + \cdots + \frac{z^k}{k a_n^k}}
\]

\[
  \text{(3)}\qquad f(z) = e^{\alpha_0 + \alpha_1 z + \cdots + \alpha_r z^r} \prod_n \Bigl(1 - \frac{z}{a_n}\Bigr)\, e^{\frac{z}{a_n} + \cdots + \frac{z^r}{r a_n^r}}
\]

\[
  \text{(4)}\qquad f(z) = C^{\mathrm{te}} \prod_n \Bigl(1 - \frac{z}{a_n}\Bigr), \quad \text{avec } \sum_n \frac{1}{|a_n|} < +\infty .
\]

\[
  \text{(5)}\qquad {\sum_n}' \frac{1}{|a_n|^{\rho+\varepsilon}} < +\infty
\]

\[
  f(z) = e^{P(z)} \prod_n \Bigl(1 - \frac{z}{a_n}\Bigr)\, e^{\frac{z}{a_n} + \cdots + \frac{z^p}{p a_n^p}}
\]

\[
  \omega - 1 \leq g \leq \omega
\]

\[
  |G(z)| \leq \bigl(K e^{-\alpha|z|}\bigr)\bigl(K' e^{\frac{\alpha}{3}|z|}\bigr)\bigl(K' e^{\frac{\alpha}{3}|z|}\bigr) = K'' e^{-\frac{\alpha}{3}|z|}
\]

\[
  n_i = \frac{n_{i-1}}{1 - \varepsilon_i} = \frac{n_{i-2}}{(1 - \varepsilon_i)(1 - \varepsilon_{i-1})} = \cdots = \frac{1}{(1 - \varepsilon_1)\cdots(1 - \varepsilon_i)} \leq \frac{1}{\prod_1^{\infty} (1 - \varepsilon_k)},
\]

\[
  \text{(1)}\qquad \sum_{\nu,\ a < r_\nu < b} \frac{1}{r_\nu^{\alpha}} = \Bigl(\frac{n(b-0)}{b^{\alpha}} - \frac{n(a+0)}{a^{\alpha}}\Bigr) + \alpha \int_a^b \frac{n(r)\,dr}{r^{\alpha+1}}
\]

\[
  \text{(1')}\qquad \sum_{\nu,\ r_\nu > a} \frac{1}{r_\nu^{\alpha}} = -\frac{n(a+0)}{a^{\alpha}} + \alpha \int_a^{\infty} \frac{n(r)\,dr}{r^{\alpha+1}}
\]

\[
  \sum_\nu \lambda(r_\nu) < +\infty \iff \int_0^{\infty} n(r)\,(-\lambda'(r))\,dr < +\infty
\]

\[
  \sum_{r_\nu > a} \lambda(r_\nu) = -\lambda(a)\,n(a+0) + \int_a^{\infty} n(r)\,(-\lambda'(r))\,dr
\]

\[
  2\pi \int_a^b n(r)\,\mu(r)\,dr = \int_a^b \int_0^{2\pi} \frac{f'(re^{i\varphi})}{f(re^{i\varphi})}\, e^{i\varphi}\, r\mu(r)\, dr\, d\varphi
\]

\[
  = \int_0^{2\pi} d\varphi \int_a^b \frac{f'(re^{i\varphi})}{f(re^{i\varphi})}\, e^{i\varphi}\, r\mu(r)\, dr = \int_0^{2\pi} d\varphi \int_a^b \frac{d}{dr}\log f(re^{i\varphi})\; r\mu(r)\, dr
\]

\[
  = \int_0^{2\pi} d\varphi \Bigl( \bigl[ r\mu(r) \log f(re^{i\varphi}) \bigr]_a^b - \int_a^b \log f(re^{i\varphi})\, d(r\mu(r)) \Bigr)
\]

\[
  \int_a^b n(r)\,\mu(r)\,dr = \bigl[ r\mu(r)\,V(r) \bigr]_a^b - \int_a^b V(r)\, d(r\mu(r))
\]

\[
  \sum_{a < r_\nu < b} \lambda(r_\nu) = \lambda(b)\,n(b-0) - \lambda(a)\,n(a+0) + \bigl[ r\mu(r)\,V(r) \bigr]_a^b + \int_a^b V(r)\, d(r\lambda'(r))
\]

\[
  \alpha_n(\lambda) = \sum_{i_1 < \cdots < i_n} \lambda_{i_1} \cdots \lambda_{i_n}
\]

\[
  \alpha_n(\lambda) \leq \frac{\|\lambda\|_1^{\,n}}{n!}
\]

\[
  \sqrt[n]{|\alpha_n(\lambda)|} = o\Bigl(\frac{1}{n}\Bigr),
\]

\[
  \alpha_n \leq \Bigl(\frac{\rho_{n+1}}{n+1} + \lambda_1\Bigr)\cdots\Bigl(\frac{\rho_{n+1}}{n+1} + \lambda_{n+1}\Bigr) e^{n+1} \leq \Bigl(\frac{\rho_1}{1} + \lambda_1\Bigr)\Bigl(\frac{\rho_2}{2} + \lambda_2\Bigr)\cdots\Bigl(\frac{\rho_{n+1}}{n+1} + \lambda_{n+1}\Bigr) e^{n+1}
\]

\[
  \alpha_n \leq \frac{\varepsilon_0\, \varepsilon_1 \cdots \varepsilon_{n+1}}{(n+1)!}\, e^{n+1}
\]

\[
  n!\,\alpha_n \leq \frac{\varepsilon_0}{n+1}\,\varepsilon_1 \cdots \varepsilon_n\, e^{n}
  \qquad
  \sqrt[n]{n!\,\alpha_n} \leq e\Bigl(\frac{\varepsilon_0}{n+1}\Bigr)^{\frac1n} (\varepsilon_1 \cdots \varepsilon_n)^{\frac1n}
\]

\[
  \alpha_n(\lambda^{(1)}, \ldots, \lambda^{(n)}) = \frac{1}{n!} \sum_{i_1, \ldots, i_n} \lambda^{(1)}_{i_1} \cdots \lambda^{(n)}_{i_n}
\]

\[
  a^{(m)}_i = \begin{cases} \dfrac{1}{m} & \text{si } i \leq m \\[4pt] 0 & \text{si } i > m \end{cases}
\]

\[
  \alpha_n(a^{(m)}) = \struck{\ill{}}\ \frac{1}{m^n}\binom{m}{n} = \frac{1}{m^n}\,\frac{m!}{n!\,(m-n)!} \quad \text{si } n \leq m,
\]

\[
  n!\,\alpha_n(a^{(m)}) = \frac{1}{m^n}\, m(m-1)\cdots(m-n+1) = \Bigl(1 - \frac{1}{m}\Bigr)\Bigl(1 - \frac{2}{m}\Bigr)\cdots\Bigl(1 - \frac{n-1}{m}\Bigr)
\]

\[
  \bigl(\|u_1 \wedge \cdots \wedge u_n\|_1\bigr)^2 \leq \alpha_n(|u_1|, \ldots, |u_n|)\; \alpha_n(|u_1|', \ldots, |u_n|')
\]

\[
  |\alpha_n(u_1, \ldots, u_n)| \leq \alpha_n(|u_1|, \ldots, |u_n|).
\]

\[
  |u_i| = \sum_\alpha \lambda^{(i)}_\alpha\, a^{(i)}_\alpha, \qquad |u_i|' = \sum_\alpha \lambda^{(i)}_\alpha\, b^{(i)}_\alpha
\]

\[
  u_1 \wedge \cdots \wedge u_n = \frac{1}{n!} \sum_{\alpha_1 \ldots \alpha_n} \lambda^{(1)}_{\alpha_1} \cdots \lambda^{(n)}_{\alpha_n}\, (a^{(1)}_{\alpha_1} \wedge \cdots \wedge a^{(n)}_{\alpha_n}) \otimes (b^{(1)}_{\alpha_1} \wedge \cdots \wedge b^{(n)}_{\alpha_n})
\]

\[
  \|u_1 \wedge \cdots \wedge u_n\|_1 \leq \frac{1}{n!} \sum_{\alpha_1 \ldots \alpha_n} \lambda^{(1)}_{\alpha_1} \cdots \lambda^{(n)}_{\alpha_n}\, \|a^{(1)}_{\alpha_1} \wedge \cdots \wedge a^{(n)}_{\alpha_n}\|\, \|b^{(1)}_{\alpha_1} \wedge \cdots \wedge b^{(n)}_{\alpha_n}\|
\]

\[
  \alpha_n(|u_1|, \ldots, |u_n|) = \frac{1}{n!} \sum_{\alpha_1 \ldots \alpha_n} \lambda^{(1)}_{\alpha_1} \cdots \lambda^{(n)}_{\alpha_n}\, \|a^{(1)}_{\alpha_1} \wedge \cdots \wedge a^{(n)}_{\alpha_n}\|^2 \struck{\ill{}}
\]

\[
  \alpha_n(|u_1|', \ldots, |u_n|') = \frac{1}{n!} \sum_{\alpha_1 \ldots \alpha_n} \lambda^{(1)}_{\alpha_1} \cdots \lambda^{(n)}_{\alpha_n}\, \|b^{(1)}_{\alpha_1} \wedge \cdots \wedge b^{(n)}_{\alpha_n}\|^2
\]

\[
  |\alpha_n(u_1, \ldots, u_n)| \leq \|u_1 \wedge \cdots \wedge u_n\|_1 \leq \tfrac12\bigl(\alpha_n(|u_1|, \ldots, |u_n|) + \alpha_n(|u_1|', \ldots, |u_n|')\bigr)
\]

\[
  |\alpha_n(A+B)|^2 \leq \alpha_n(|A| + |B|)\; \alpha_n(|A|' + |B|')
\]

\begin{gather*}
  |\alpha_n(A+B)|^2 = \Bigl|\sum_k C_n^k\, \alpha_n(\overbrace{A, \ldots, A}^{k}, \overbrace{B, \ldots, B}^{n-k})\Bigr|^2 \\
  \leq \Bigl(\sum_k C_n^k\, \alpha_n(\overbrace{|A|, \ldots, |A|}^{k}, \overbrace{|B|, \ldots, |B|}^{n-k})^{\frac12}\, \alpha_n(\overbrace{|A|', \ldots, |A|'}^{k}, |B|', \ldots)^{\frac12}\Bigr)^2
\end{gather*}

\[
  |\alpha_n(A+B)|^2 \leq \Bigl(\sum_k C_n^k\, \alpha_n(|A|, \ldots, |A|, |B|, \ldots, |B|)\Bigr)^{\frac12} \Bigl(\sum_k \ldots\Bigr)^{\frac12}
\]

\[
  = \bigl(\alpha_n(|A| + |B|)\bigr)^{\frac12} \bigl(\alpha_n(|A|' + |B|')\bigr)^{\frac12}
\]

\[
  |\det(1 + A + B)| \leq \tfrac12\bigl(\det(1 + |A| + |B|) + \det(1 + |A|' + |B|')\bigr)
\]

\[
  |\det(1 + A + B)| \leq \det(1 + |A|)\det(1 + |B|)
\]

\[
  \det(1 + \uncertain{H} + K) = \det(1 + \uncertain{L})\Bigl(1 + \frac{K}{1 + L}\Bigr) = \det(1 + L)\det(1 + HK)
\]

\[
  \Lambda^n HK = (\Lambda^n H)(\Lambda^n K), \quad \text{d'où} \quad \|\Lambda^n HK\|_1 \leq \|\Lambda^n H\|_\infty\, \|\Lambda^n K\|_1 \quad \text{et} \quad \|\Lambda^n H\|_\infty \leq \|H\|_\infty^{\,n} \leq 1.
\]

\[
  \det(1 + |A + B|) \leq \det(1 + |A|)\det(1 + |B|)
\]

\[
  \det(1 + UA + UB) \leq \det(1 + |UA|)\det(1 + |UB|) = \det(1 + |A|)\det(1 + |B|)
\]

\begin{gather*}
  (1)\qquad \|A\|_p = \|A^*\|_p \qquad \text{pour } A \in \mathcal{L}^p(H),\ B \in \mathcal{L}(H) \\
  (2)\qquad \|BA\|_p \leq \|B\|\,\|A\|_p, \qquad \|AB\|_p \leq \|B\|\,\|A\|_p
\end{gather*}

\[
  (3)\qquad \|A + B\|_p \leq \|A\|_p + \|B\|_p, \qquad \|\lambda A\|_p = |\lambda|\,\|A\|_p \qquad (1 \leq p \leq \infty)
\]

\[
  (4)\qquad S_p(A + B) \leq S_p(A) + S_p(B) \qquad (0 < p \leq 1)
\]

\[
  (5)\qquad \sum |\lambda_i|^p \leq S_p(A)
\]

\[
  (6)\qquad \|AB\|_r \leq \|A\|_p\,\|B\|_q
\]

\[
  |\operatorname{Tr}(AB)| \leq \|A\|_p, \qquad \text{et} \qquad \operatorname*{Sup}_{\|B\|_{p'} \leq 1} |\operatorname{Tr} AB| \leq \|A\|_p
\]

\[
  (7)\qquad \|A\|_p = \operatorname*{Sup}_{\|B\|_{p'} \leq 1} |\operatorname{Tr} AB|
\]

\[
  (8)\qquad \sum \alpha_i^{\,p} = \frac{p \sin \pi p}{\pi} \int_0^\infty \frac{\log f(v)}{v^{1+p}}\, dv
\]

\[
  (9)\qquad S_p(A) = \frac{p \sin \pi p}{\pi} \int_0^\infty \frac{\log M_{|A|}(v)}{v^{1+p}}\, dv
\]

\[
  S_p(A + B) = S_p(|A + B|) = \frac{p \sin \pi p}{\pi} \int_0^\infty \frac{\log M_{|A+B|}(v)}{v^{1+p}}\, dv
\]

\[
  (10)\qquad M_{|A+B|}(v) \leq M_{|A|}(v) \cdot M_{|B|}(v) \qquad \text{d'où}
\]

\[
  S_p(A + B) \leq \frac{p \sin \pi p}{\pi} \int_0^\infty \frac{\log M_{|A|}(v)}{v^{1+p}}\, dv + \frac{p \sin \pi p}{\pi} \int_0^\infty \frac{\log M_{|B|}(v)}{v^{1+p}}\, dv
\]

\[
  s_p(A) = \sum |\lambda_i|^p.
\]

\[
  f(z) = \det(1 - zA) \qquad (0 < p < 1)
\]

\[
  \struck{(10)}\qquad s_p(A) \leq p^2 \int_0^\infty \frac{\log M_A(r)}{r^{p+1}}\, dr
\]

\[
  s_p(A) \leq p^2 \int_0^\infty \frac{\log M_{|A|}(v)}{v^{p+1}}\, dv
\]

\[
  (10)\qquad s_p(A) \leq \frac{p\pi}{\sin p\pi}\, S_p(A)
\]

\[
  s_p(A) \leq \frac{\frac{p}{n}\pi}{\sin \frac{p}{n}\pi}\, S_{p/n}(A^n)
\]

\[
  S_{p/n}(A^n) = \bigl(\|A^n\|_{p/n}\bigr)^{\frac{p}{n}} \leq \|A\|_p^{\,p} = S_p(A)
\]

\[
  s_p(A) \leq \frac{\pi \frac{p}{n}}{\sin \pi \frac{p}{n}}\, S_p(A).
\]

\[
  S_r(AB) = S_{r/2}\bigl(AB(AB)^*\bigr) = \struck{S_{r/2m}\bigl((AB(AB)^*)^m\bigr)}
\]

\[
  S_r(AB) = S_{r/2}(L) = S_{r/2m}(L^m), \quad \text{d'où, en vertu de la formule (9),} \qquad k(\alpha) = \frac{\pi\alpha}{\sin \pi\alpha}
\]

\[
  S_r(AB) = S_{r'}(L^m) = \frac{r' \sin \pi r'}{\pi} \int_0^\infty \frac{\log M_{L^m}(v)}{v^{1+r'}}\, dv
\]

\[
  (11)\qquad \|A_1 A_2 \cdots A_{2^n}\|_1 \leq \|A_1\|_{2^n}\, \|A_2\|_{2^n} \cdots \|A_{2^n}\|_{2^n}
\]

\[
  \|AB\|_{2^{n-1}} = \Bigl(\bigl\|\bigl(AB(AB)^*\bigr)^{\frac{2^{n-1}}{2}}\bigr\|_1\Bigr)^{\frac{1}{2^{n-1}}} = \bigl(\|C\|_1\bigr)^{\frac{1}{2^{n-1}}}
\]

\[
  C = (ABB^*A^*)^{2^{n-2}} = A(BB^*)(A^*A)\cdots(BB^*)A^*
\]

\[
  \|C\|_1 = \operatorname{Tr} C = \operatorname{Tr} C' \quad \text{où } C' = A^*A \cdot BB^* \cdot A^*A \cdots BB^*
\]

\[
  \|C'\|_1 \leq \|A^*A\|_{2^{n-1}}\, \|BB^*\|_{2^{n-1}} \cdots \|BB^*\|_{2^{n-1}} = \bigl(\|A^*A\|_{2^{n-1}}\bigr)^{2^{n-2}} \bigl(\|B^*B\|_{2^{n-1}}\bigr)^{2^{n-2}}
\]

\[
  \|AB\|_{2^{n-1}} = \bigl(\|C\|_1\bigr)^{\frac{1}{2^{n-1}}} \leq \bigl(\|C'\|_1\bigr)^{\frac{1}{2^{n-1}}} \leq \|A\|_{2^n}\, \|B\|_{2^n},
\]

\[
  \|h^\alpha k^\beta\|_1 \leq \underbrace{\|h\|_{2^n} \cdots \|h\|_{2^n}}_{\alpha} \underbrace{\|k\|_{2^n} \cdots \|k\|_{2^n}}_{\beta} = \bigl(\operatorname{Tr} h^{2^n}\bigr)^{\frac{\alpha}{2^n}} \bigl(\operatorname{Tr}(k^{2^n})\bigr)^{\frac{\beta}{2^n}}
\]

\[
  \|AB\|_1 \leq \bigl(\operatorname{Tr} H^{\lambda}\bigr)^{\frac1\lambda} \bigl(\operatorname{Tr} K^{\mu}\bigr)^{\frac1\mu} \qquad \text{où } \lambda = \frac{2^n}{\alpha},\ \mu = \frac{2^n}{\beta}
\]

\[
  \|AB\|_1 \leq \bigl(\operatorname{Tr} H^p\bigr)^{\frac1p} \bigl(\operatorname{Tr} H^{p'}\bigr)^{\frac{1}{p'}}
\]

\[
  \|A\|_p = \operatorname*{Sup}_{\|B\|_{p'} \leq 1} |\operatorname{Tr} AB| \qquad (\text{d'où} \ \ill{})
\]

\[
  \|A_1 + A_2\|_p \leq \|A_1\|_p + \|A_2\|_p \qquad ) \ \uncertain{de} \ \uncertain{même}
\]

\[
  (12)\qquad \alpha_n(L^m) \leq \alpha_n(H^m)\, \alpha_n(K^m)
\]

\begin{gather*}
  \struck{H = A^*A,\ K = B^*B,\ L = (AB)^*(AB)} \\
  \struck{\operatorname{Tr} L^m \leq \operatorname{Tr} H^m\, \operatorname{Tr} K^m} \\
  \struck{(\|L\|_m)^m \leq (\|H\|_m)^m (\|K\|_m)^m} \\
  \struck{\|AB\|_m \leq \|A\|_m \|B\|_m.\ \text{Or, on a}\ \ill{}} \\
  \struck{\ill{}\ \|AB\|_m \leq \|A\|_{\frac m2} \|B\|_{\frac m2}}
\end{gather*}

\[
  \operatorname{Tr}\bigl[(ABB^*A^*) \cdots (ABB^*A^*)\bigr] \leq \bigl(\|A^*A\|_m\bigr)^m \bigl(\|B^*B\|_m\bigr)^m
\]

\[
  \bigl(\|A^*A\|_{2m}\bigr)^m \bigl(\|BB^*\|_{2m}\bigr)^m
\]

\[
  S_r(AB) = S_{r/2}(L) = S_{r/2m}(L^m) \struck{\ill{}}
\]

\[
  (11)\qquad S_r(AB) = S_{r/2m}(L^m) = \frac{r/2m}{\lambda(r/2m)} \int_0^\infty \frac{\log M_{L^m}(\rho)}{\rho^{1 + r/2m}}\, d\rho
\]

\[
  M_{L^m}(\rho) \leq \sum \alpha_n(H^m)\, \alpha_n(K^m)\, \rho^n
\]

\[
  M_{L^m}(\rho) \leq \sum \alpha_n(H^m)\, \alpha_n(K^m)\, \rho^{np'r + nq'r} = \sum_n \bigl(\alpha_n(H^m)\rho^{np'r}\bigr)\bigl(\alpha_n(K^m)\rho^{nq'r}\bigr)
\]

\[
  \leq \Bigl(\sum_n \alpha_n(H^m)\, \rho^{np'r}\Bigr)\Bigl(\sum_n \alpha_n(K^m)\, \rho^{nq'r}\Bigr) \struck{\leq} = M_{H^m}(\rho^{p'r})\, M_{H}(\rho^{q'r})
\]

\[
  (13)\qquad \frac{\lambda(r/2m)}{r/2m}\, S_r(AB) \leq \int_0^\infty \frac{\log M_{H^m}(\rho^{p'r})}{\rho^{1 + r/2m}}\, d\rho + \int_0^\infty \frac{\log M_{K^m}(\rho^{q'r})}{\rho^{1 + r/2m}}\, d\rho
\]

\[
  \int_0^\infty \frac{\log M_{H^m}(\rho^{p'r})}{\rho^{r/2m}}\, \frac{d\rho}{\rho} = pr' \int_0^\infty \frac{\log M_{H^m}(\sigma)}{\sigma^{p/2m}}\, \frac{d\sigma}{\sigma}
\]

\[
  pr'\, \frac{\lambda(p/2m)}{p/2m}\, S_{p/2m}(H^m) = \frac{\lambda(p/2m)}{r/2m}\, S_p(A).
\]

\[
  \lambda(r/2m)\, S_r(AB) \leq \lambda(p/2m)\, S_p(A) + \lambda(q/2m)\, S_q(B)
\]

\[
  S_r(AB) \leq \frac{\sin \pi \frac{r}{2m}}{\sin \pi \frac{p}{2m}}\, S_p(A) + \frac{\sin \pi \frac{r}{2m}}{\sin \pi \frac{q}{2m}}\, S_q(B)
\]

\[
  (14)\qquad S_r(AB) \leq \frac{r}{p}\, S_p(A) + \frac{r}{q}\, S_q(B)
\]

\[
  \struck{\ill{}}\qquad (\lambda\mu)^r\, S_r(AB) \leq \frac{\lambda^p}{p}\, S_p(A) + \frac{\mu^q}{q}\, S_q(B)
\]

\[
  a = S_p(A), \quad b = S_q(B), \quad c = S_r(C), \quad \ill{}
\]

\[
  c \leq \operatorname*{Inf}_{\substack{\lambda \geq 0 \\ \mu \geq 0}} \frac{a\,\dfrac{\lambda^p}{p} + b\,\dfrac{\mu^q}{q}}{\struck{\ill{}}\,(\lambda\mu)^r}
\]

\[
  c \leq \operatorname*{Inf}_{\substack{x \geq 0 \\ y \geq 0}} \frac{\alpha a\, x + \beta b\, y}{x^\alpha y^\beta} = \operatorname*{Inf}_{\substack{x \geq 0 \\ y \geq 0 \\ x^\alpha y^\beta = 1}} \alpha a\, x + \beta b\, y
\]

\[
  \frac{\alpha a}{\alpha x^{\alpha-1} y} = \frac{\beta b}{\beta x^\alpha y^{\beta-1}}
\]

\[
  \bigl\|\,|H^r - K^r|^{1/r}\,\bigr\| \leq \|H - K\|
\]

\begin{gather*}
  \rho_{m+n-1}(vu) \leq \rho_m(u)\, \rho_n(v) \\
  \rho_{m+n-1}(u+v) \leq \rho_m(u) + \rho_n(v)
\end{gather*}

\[
  \operatorname*{Sup}_{x \perp H \wedge E_{n-1},\ \|x\| \leq 1} \|vux\| \leq \rho_n(u)\, \rho_m(v)
\]

\[
  \rho_p(vu) \leq \operatorname{Inf} \operatorname*{Sup}_{x \perp E_{p-1},\ \|x\| \leq 1} \|vux\| \leq \rho_n(u)\, \rho_m(v)
\]

\[
  \bigl\|\,|H^r - K^r|^{1/r}\,\bigr\| \leq \|H - K\| \qquad \text{pour } 0 < r < 1 .
\]

\[
  S_{p/r}(H^r - K^r) = S_p\bigl(|H^r - K^r|^{1/r}\bigr)
\]

\[
  \sum_{r_\nu(A_i) \leq \rho} \bigl(r_\nu(A_i)\bigr)^p \leq \varepsilon
\]

\[
  \omega(t) = p\rho^p\,(t - \log \rho) + \rho^p = \rho^p \log e\Bigl(\frac{e^t}{\rho}\Bigr)^p
\]

\[
  \sum_{r_\nu \leq \rho} r_\nu^{\,p} + \rho^p \sum_{r_\nu > \rho} \log e\Bigl(\frac{r_\nu}{\rho}\Bigr)^p
  \leq
  \sum_{s_\nu \leq \rho} s_\nu^{\,p} + \rho^p \sum_{s_\nu > \rho} \log e\Bigl(\frac{s_\nu}{\rho}\Bigr)^p
  \qquad \text{(ou } |A_i| \to |A|\text{)}
\]

\[
  \sum_{r_\nu(A_i) \leq \rho} r_\nu(A_i)^p + \rho^p \sum_{r_\nu(A_i) > \rho} \log e\Bigl(\frac{r_\nu(A_i)}{\rho}\Bigr)^p \leq \varepsilon
\]

\begin{align*}
  \rho^p \sum_{r_\nu > \rho} \log e\Bigl(\frac{r_\nu}{\rho}\Bigr)^p
  &= \rho^p \int_{r > \rho}^{+\infty} - \log e\Bigl(\frac{r}{\rho}\Bigr)^p\, dn(r) \\
  &= \rho^p \Bigl[- n(r) \log e\Bigl(\frac{r}{\rho}\Bigr)^p\Bigr]_{\rho+0}^{+\infty} + p\rho^p \int_\rho^{+\infty} \frac{n(r)}{r}\, dr \\
  &= \rho^p\, n(\rho) + p\rho^p \int_\rho^{+\infty} \frac{n(r)}{r}\, dr
\end{align*}

\[
  p\rho^p \int_\rho^{+\infty} \frac{n(r)}{r}\, dr < p\rho^p \int_{\uncertain{0}}^{+\infty} \frac{n(r)}{r}\, dr \to 0 \quad \text{si } \rho \to 0 .
\]

\[
  \sum_{r_\nu(A_i) \leq \rho} \bigl(r_\nu(A_i)\bigr)^p = S_p(A_i) - \sum_{r_\nu(A_i) > \rho} \bigl(r_\nu(A_i)\bigr)^p
  \to \struck{\leq}\; S_p(A) - \sum_{r_\nu > \rho} r_\nu^{\,p} = \sum_{r_\nu \leq \rho} r_\nu^{\,p}
\]

\[
  \frac{1}{1+u} = 1 - u\, \frac{R(u)}{\det(1+u)} \qquad R(u) = \sum_0^\infty P_k(u) = \frac{\det(1+u)}{1+u}
\]

\[
  \text{(*), (**)}\qquad \struck{\ill{}}\quad |P_n(u)| \leq P_n(|u|),\qquad |R(u)| \leq R(|u|)
\]

\[
  \text{(**)}\qquad \struck{\ill{}}\quad 0 \leq P_n(u) \leq \alpha_n(u) \cdot 1 \quad \text{d'où} \quad \struck{\ill{}}\quad 0 \leq R(u) \leq \det(1+u)
\]

\[
  \text{(***)}\qquad
  \begin{cases}
    \|R(u)\|_\infty \leq \|R(|u|)\|_\infty \leq \det(1+|u|) \leq e^{\|u\|_1} \\[4pt]
    \|P_n(u)\|_\infty \leq \|P_n(|u|)\|_\infty \leq \alpha_n(|u|) \leq \dfrac{\|u\|_1^{\,n}}{n!}
  \end{cases}
\]

\[
  p_i(u) = \sum_{(I)}{}' \lambda_{i_1} \cdots \lambda_{i_n},
\]

\[
  \sum_{k=0}^{n} u^{n-k}\, \alpha_k(u)\, (-1)^k = (-1)^n \sum{}' \lambda_{i_1} \cdots \lambda_{i_n} \ \ill{}
\]

\[
  \|R(u)\|_\infty \geq |\det(1+u)| \qquad (\ill{})
\]