From e36b50510d61beafbed644d6654793cdc7ef2faf Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 20 Apr 2023 20:07:45 +0200 Subject: [PATCH] feat(geometria): aggiunge norme e distanze euclidee e complementi sui gruppi matriciali particolari --- .../main.pdf | Bin .../main.tex | 0 .../main.pdf | Bin .../main.tex | 0 .../main.pdf | Bin 267945 -> 291373 bytes .../main.tex | 210 +++++++++++++++--- 6 files changed, 178 insertions(+), 32 deletions(-) rename Geometria 1/{Prodotto scalare => Prodotto scalare e hermitiano}/2023-03-22, Introduzione al prodotto scalare/main.pdf (100%) rename Geometria 1/{Prodotto scalare => Prodotto scalare e hermitiano}/2023-03-22, Introduzione al prodotto scalare/main.tex (100%) rename Geometria 1/{Prodotto scalare => Prodotto scalare e hermitiano}/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf (100%) rename Geometria 1/{Prodotto scalare => Prodotto scalare e hermitiano}/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.tex (100%) rename Geometria 1/{Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani => Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale}/main.pdf (57%) rename Geometria 1/{Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani => Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale}/main.tex (62%) diff --git a/Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/2023-03-22, Introduzione al prodotto scalare/main.pdf similarity index 100% rename from Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.pdf rename to Geometria 1/Prodotto scalare e hermitiano/2023-03-22, Introduzione al prodotto scalare/main.pdf diff --git a/Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.tex b/Geometria 1/Prodotto scalare e hermitiano/2023-03-22, Introduzione al prodotto scalare/main.tex similarity index 100% rename from Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.tex rename to Geometria 1/Prodotto scalare e hermitiano/2023-03-22, Introduzione al prodotto scalare/main.tex diff --git a/Geometria 1/Prodotto scalare/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf similarity index 100% rename from Geometria 1/Prodotto scalare/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf rename to Geometria 1/Prodotto scalare e hermitiano/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf diff --git a/Geometria 1/Prodotto scalare/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.tex b/Geometria 1/Prodotto scalare e hermitiano/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.tex similarity 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G!T$q9FqXam diff --git a/Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani/main.tex b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex similarity index 62% rename from Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani/main.tex rename to Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex index 5cc1cc7..7104bf4 100644 --- a/Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani/main.tex +++ b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex @@ -11,7 +11,7 @@ \maketitle \begin{center} - \Large \textbf{Prodotti hermitiani e teorema spettrale} + \Large \textbf{Prodotti hermitiani, spazi euclidei e teorema spettrale} \end{center} \begin{note} @@ -79,7 +79,8 @@ \begin{itemize} \item $(A + B)^* = A^* + B^*$, - \item $(AB)^* = B^* A^*$. + \item $(AB)^* = B^* A^*$, + \item $(A\inv)^* = (A^*)\inv$, se $A$ è invertibile. \end{itemize} \end{remark} @@ -352,24 +353,60 @@ D'ora in poi, nel corso del documento, s'intenderà per $\varphi$ un prodotto scalare (o eventualmente hermitiano) di $V$. \end{note} - \begin{definition} + \begin{definition} (operatori simmetrici) Sia $f \in \End(V)$. Si dice allora che $f$ è \textbf{simmetrico} se $f = f^\top$. \end{definition} - \begin{definition} + \begin{definition} (applicazioni e matrici ortogonali) Sia $f \in \End(V)$. Si dice allora che $f$ è \textbf{ortogonale} se $\varphi(\v, \w) = \varphi(f(\v), f(\w))$. + Sia $A \in M(n, \KK)$. Si dice dunque che $A$ è \textbf{ortogonale} se $A^\top A = A A^\top = I_n$. \end{definition} \begin{definition} + Le matrici ortogonali di $M(n, \KK)$ formano un sottogruppo moltiplicativo di $\GL(n, \KK)$, detto \textbf{gruppo ortogonale}, + e indicato con $O_n$. Il sottogruppo di $O_n$ contenente solo le matrici con determinante pari a $1$ è + detto \textbf{gruppo ortogonale speciale}, e si denota con $SO_n$. + \end{definition} + + \begin{remark} + Si possono classificare in modo semplice alcuni di questi gruppi ortogonali per $\KK = \RR$. \\ + + \li $A \in O_n \implies 1 = \det(I_n) = \det(A A^\top) = \det(A)^2 \implies \det(A) = \pm 1$. + \li $A = (a) \in O_1 \iff A^\top A = I_1 \iff a^2 = 1 \iff a = \pm 1$, da cui si ricava che l'unica matrice + di $SO_1$ è $(1)$. Si osserva inoltre che $O_1$ è abeliano di ordine $2$, e quindi che $O_1 \cong \ZZ/2\ZZ$. \\ + \li $A = \Matrix{a & b \\ c & d} \in O_2 \iff \Matrix{a^2 + b^2 & ab + cd \\ ab + cd & c^2 + d^2} = A^\top A = I_2 \iff \system{a^2 + b^2 = c^2 + d^2 = 1, \\ ab + cd = 0.}$ \\ + + Si ricava pertanto che si può identificare + $A$ con le funzioni trigonometriche $\cos(\theta)$ e $\sin(\theta)$ nelle due forme: + \begin{align*} + &A = \Matrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)} \quad &\text{(}\!\det(A) = 1, A \in SO_2\text{)}, \\ + &A = \Matrix{\cos(\theta) & \sin(\theta) \\ \sin(\theta) & -\cos(\theta)} \quad &\text{(}\!\det(A) = -1\text{)}. + \end{align*} + \end{remark} + + \begin{definition} (applicazioni e matrici hermitiane) Sia $f \in \End(V)$ e si consideri il prodotto hermitiano $\varphi$. Si dice allora che - $f$ è \textbf{hermitiano} se $f = f^*$. + $f$ è \textbf{hermitiano} se $f = f^*$. Sia $A \in M(n, \CC)$. Si dice dunque che $A$ + è \textbf{hermitiana} se $A = A^*$. \end{definition} - \begin{definition} + \begin{definition} (applicazioni e matrici unitarie) Sia $f \in \End(V)$ e si consideri il prodotto hermitiano $\varphi$. Si dice allora che - $f$ è \textbf{unitario} se $\varphi(\v, \w) = \varphi(f(\v), f(\w))$. + $f$ è \textbf{unitario} se $\varphi(\v, \w) = \varphi(f(\v), f(\w))$. Sia $A \in M(n, \CC)$. + Si dice dunque che $A$ è \textbf{unitaria} se $A^* A = A A^* = I_n$. + \end{definition} + + \begin{definition} + Le matrici unitarie di $M(n, \CC)$ formano un sottogruppo moltiplicativo di $\GL(n, \CC)$, detto \textbf{gruppo unitario}, + e indicato con $U_n$. Il sottogruppo di $U_n$ contenente solo le matrici con determinante pari a $1$ è + detto \textbf{gruppo unitario speciale}, e si denota con $SU_n$. \end{definition} + \begin{remark}\nl + \li $A \in U_n \implies 1 = \det(I_n) = \det(A A^*) = \det(A) \conj{\det(A)} = \abs{\det(A)}^2 = 1$. + \li $A = (a) \in U_1 \iff A^* A = I_1 \iff \abs{a}^2 = 1 \iff a = e^{i\theta}$, $\theta \in [0, 2\pi)$, ossia il numero complesso $a$ appartiene alla circonferenza di raggio unitario. + \end{remark} + \begin{definition} (spazio euclideo reale) Si definisce \textbf{spazio euclideo reale} uno spazio vettoriale $V$ su $\RR$ dotato del prodotto scalare standard $\varphi = \innprod{\cdot, \cdot}$. @@ -386,7 +423,7 @@ \end{definition} \begin{proposition} - Sia $(V, \varphi)$ uno spazio euclideo reale e sia $\basis$ una base ortonormale di $V$. Allora $f \in \End(V)$ è simmetrico $\iff$ $M_\basis(f) = M_\basis(f)^\top$. + Sia $(V, \varphi)$ uno spazio euclideo reale e sia $\basis$ una base ortonormale di $V$. Allora $f \in \End(V)$ è simmetrico $\iff$ $M_\basis(f) = M_\basis(f)^\top$ $\iff$ $M_\basis(f)$ è simmetrica. \end{proposition} \begin{proof} @@ -396,54 +433,163 @@ Se invece $M_\basis(f)^\top = M_\basis(f)$, $\varphi(\v, f(\w)) =$ $[\v]_\basis^\top M_\basis(\varphi) (M_\basis(f) [\w]_\basis)$ $= [\v]_\basis^\top M_\basis(f) [\w]_\basis$ $= [\v]_\basis^\top M_\basis(f)^\top [\w]_\basis = (M_\basis(f) [\v]_\basis)^\top [\w]_\basis = (M_\basis(f) [\v]_\basis)^\top M_\basis(\varphi) [\w]_\basis = \varphi(f(\v), \w)$, e quindi $f$ è simmetrico. \end{proof} - % TODO: aggiungere matrici ortogonali, hermitiani e unitari. + \begin{proposition} + Sia $(V, \varphi)$ uno spazio euclideo reale e sia $\basis$ una base ortonormale di $V$. Allora + $f \in \End(V)$ è ortogonale $\iff$ $M_\basis(f) M_\basis(f)^\top = M_\basis(f)^\top M_\basis(f) = I_n$ $\defiff$ $M_\basis(f)$ è ortogonale. + \end{proposition} + + \begin{proof} + Si osserva che $M_\basis(\varphi) = I_n$. Sia $\basis = \{ \vv 1, \ldots, \vv n\}$. Se $f$ è ortogonale, allora + $[\v]_\basis^\top \, [\w]_\basis = [\v]_\basis^\top \, M_\basis(\varphi) [\w]_\basis = \varphi(\v, \w) = + \varphi(f(\v), f(\w)) = (M_\basis(f) [\v]_\basis)^\top \, M_\basis(\varphi) (M_\basis(f) [\w]_\basis) = + [\v]_\basis^\top M_\basis(f)^\top M_\basis(\varphi) M_\basis(f) [\w]_\basis = [\v]_\basis^\top M_\basis(f)^\top M_\basis(f) [\w]_\basis$. Allora, come visto nella proposizione precedente, si ricava che $M_\basis(f)^\top M_\basis(f) = I_n$. Dal momento che gli inversi sinistri sono anche inversi destri, $M_\basis(f)^\top M_\basis(f) = M_\basis(f) M_\basis(f)^\top = I_n$. \\ + + Se invece $M_\basis(f)^\top M_\basis(f) = M_\basis(f) M_\basis(f)^\top = I_n$, $\varphi(\v, \w) = [\v]_\basis^\top [\w]_\basis = [\v]_\basis^\top M_\basis(f)^\top M_\basis(f) [\w]_\basis = + (M_\basis(f) [\v]_\basis)^\top (M_\basis(f) [\w]_\basis) =$ $(M_\basis(f) [\v]_\basis)^\top M_\basis(\varphi) (M_\basis(f) [\w]_\basis) = \varphi(f(\v), f(\w))$, e quindi + $f$ è ortogonale. + \end{proof} - \begin{proposition} Sia $\basis$ una base di $V$. Allora - $f \in \End(V)$ è simmetrico $\iff$ $M_\basis(f)$ è una matrice simmetrica. + \begin{proposition} + Sia $(V, \varphi)$ uno spazio euclideo complesso e sia $\basis$ una base ortonormale di $V$. Allora $f \in \End(V)$ è hermitiano $\iff$ $M_\basis(f) = M_\basis(f)^*$ $\defiff$ $M_\basis(f)$ è hermitiana. \end{proposition} - \begin{proof} Si dimostrano le due implicazioni separatamente. \\ + \begin{proof} + Si osserva che $M_\basis(\varphi) = I_n$. Sia $\basis = \{ \vv 1, \ldots, \vv n\}$. Se $f$ è hermitiano, allora $[\v]_\basis^* \, M_\basis(f) [\w]_\basis = [\v]_\basis^* M_\basis(\varphi) (M_\basis(f) [\w]_\basis) = \varphi(\v, f(\w)) = \varphi(f(\v), \w) = (M_\basis(f) [\v]_\basis)^* M_\basis(\varphi) [\w]_\basis = [\v]_\basis^* \, M_\basis(f)^* [\w]_\basis$. Allora, come visto nella proposizione precedente, si ricava che $M_\basis(f) = M_\basis(f)^*$. - \rightproof Se $f$ è simmetrico, $\varphi(\v, f(\w)) = \varphi(f(\v), \w)$. Quindi - $[\v]_\basis M_\basis(\varphi)$ + Se invece $M_\basis(f)^* = M_\basis(f)$, $\varphi(\v, f(\w)) =$ $[\v]_\basis^* M_\basis(\varphi) (M_\basis(f) [\w]_\basis)$ $= [\v]_\basis^* M_\basis(f) [\w]_\basis$ $= [\v]_\basis^* M_\basis(f)^* [\w]_\basis = (M_\basis(f) [\v]_\basis)^* [\w]_\basis = (M_\basis(f) [\v]_\basis)^* M_\basis(\varphi) [\w]_\basis = \varphi(f(\v), \w)$, e quindi $f$ è hermitiano. \end{proof} - \hr - \begin{proposition} - Se $V = \RR^n$ con prodotto canonico $\varphi(\vec x, \vec y) = \vec x ^\top \vec y$. Sono allora equivalenti i seguenti fatti: + Sia $(V, \varphi)$ uno spazio euclideo complesso e sia $\basis$ una base ortonormale di $V$. Allora $f \in \End(V)$ è unitario $\iff$ $M_\basis(f) M_\basis(f)^* = M_\basis(f)^* M_\basis(f) = I_n$ $\defiff$ $M_\basis(f)$ è unitaria. + \end{proposition} + + \begin{proof} + Si osserva che $M_\basis(\varphi) = I_n$. Sia $\basis = \{ \vv 1, \ldots, \vv n\}$. Se $f$ è unitario, allora + $[\v]_\basis^* \, [\w]_\basis = [\v]_\basis^* \, M_\basis(\varphi) [\w]_\basis = \varphi(\v, \w) = + \varphi(f(\v), f(\w)) = (M_\basis(f) [\v]_\basis)^* \, M_\basis(\varphi) (M_\basis(f) [\w]_\basis) = + [\v]_\basis^* M_\basis(f)^* M_\basis(\varphi) M_\basis(f) [\w]_\basis = [\v]_\basis^* M_\basis(f)^* M_\basis(f) [\w]_\basis$. Allora, come visto nella proposizione precedente, si ricava che $M_\basis(f)^* M_\basis(f) = I_n$. Dal momento che gli inversi sinistri sono anche inversi destri, $M_\basis(f)^* M_\basis(f) = M_\basis(f) M_\basis(f)^* = I_n$. \\ + + Se invece $M_\basis(f)^* M_\basis(f) = M_\basis(f) M_\basis(f)^* = I_n$, $\varphi(\v, \w) = [\v]_\basis^* [\w]_\basis = [\v]_\basis^* M_\basis(f)^* M_\basis(f) [\w]_\basis$ $= + (M_\basis(f) [\v]_\basis)^* (M_\basis(f) [\w]_\basis)$ $= (M_\basis(f) [\v]_\basis)^* M_\basis(\varphi) (M_\basis(f) [\w]_\basis) = \varphi(f(\v), f(\w))$, e quindi + $f$ è unitario. + \end{proof} + + \begin{proposition} + Sia $V = \RR^n$ uno spazio vettoriale col prodotto scalare standard $\varphi$. Allora sono equivalenti i seguenti fatti: \begin{enumerate}[(i)] \item $A \in O_n$, - \item $f_A : V \to V$ con $f_A(\vec x) = A \vec x$ è ortogonale, - \item Le colonne (e le righe) di $A$ formano una base ortonormale di $V$. + \item $f_A$ è un operatore ortogonale, + \item le colonne e le righe di $A$ formano una base ortonormale di $V$. \end{enumerate} \end{proposition} \begin{proof} - (1 - 2) ovvio - (2 - 3) $f_A$ manda basi ortonormali in basi ortonormali, e quindi - così sono ortonormali le colonne di $A$. Analogamente per le righe - considerando $A^\top A = I$. - (3 - 1) $A^\top A = I$. + Sia $\basis$ la base canonica di $V$. Allora $M_\basis(f_A) = A$, e quindi, per una proposizione + precedente, $f_A$ è un operatore ortogonale. Viceversa si deduce che se $f_A$ è un operatore ortogonale, + $A \in O_n$. Dunque è sufficiente dimostrare che $A \in O_n \iff$ le colonne e le righe di $A$ formano una + base ortonormale di $V$. \\ + + \rightproof Se $A \in O_n$, in particolare $A \in \GL(n, \RR)$, e quindi $A$ è invertibile. Allora le + sue colonne e le sue righe formano già una base di $V$, essendo $n$ vettori di $V$ linearmente indipendenti. + Inoltre, poiché $A \in O_n$, $\varphi(\e i, \e j) = \varphi(A \e i, A \e j)$, e quindi le colonne di $A$ si mantengono a due a due ortogonali tra di loro, mentre $\varphi(A \e i, A \e i) = \varphi(\e i, \e i) = 1$. + Pertanto le colonne di $A$ formano una base ortonormale di $V$. \\ + + Si osserva che anche $A^\top \in O_n$. Allora le righe di $A$, che non sono altro che + le colonne di $A^\top$, formano anch'esse una base ortonormale di $V$. \\ + + \leftproof Nel moltiplicare $A^\top$ con $A$ altro non si sta facendo che calcolare il prodotto + scalare $\varphi$ tra ogni riga di $A^\top$ e ogni colonna di $A$ , ossia $(A^* A)_{ij} = \varphi((A^\top)_i, A^j) = \varphi(A^i, A^j) = \delta_{ij}$. + Quindi $A^\top A = A A^\top = I_n$, da cui si deduce che $A \in O_n$. \end{proof} - + \begin{proposition} - Se $V = \CC^n$ con prodotto canonico hermitiano, sono equivalenti - i seguenti fatti: + Sia $V = \CC^n$ uno spazio vettoriale col prodotto hermitiano standard $\varphi$. Allora sono equivalenti i seguenti fatti: \begin{enumerate}[(i)] \item $A \in U_n$, - \item $f_A : V \to V$ con $f_A(\vec x) = A \vec x$ è unitaria, - \item Le colonne (e le righe) di $A$ formano una base ortonormale - di $V$. + \item $f_A$ è un operatore unitario, + \item le colonne e le righe di $A$ formano una base ortonormale di $V$. \end{enumerate} \end{proposition} - + \begin{proof} - Come prima. + Sia $\basis$ la base canonica di $V$. Allora $M_\basis(f_A) = A$, e quindi, per una proposizione + precedente, $f_A$ è un operatore unitario. Viceversa si deduce che se $f_A$ è un operatore unitario, + $A \in U_n$. Dunque è sufficiente dimostrare che $A \in U_n \iff$ le colonne e le righe di $A$ formano una + base ortonormale di $V$. \\ + + \rightproof Se $A \in U_n$, in particolare $A \in \GL(n, \RR)$, e quindi $A$ è invertibile. Allora le + sue colonne e le sue righe formano già una base di $V$, essendo $n$ vettori di $V$ linearmente indipendenti. + Inoltre, poiché $A \in U_n$, $\varphi(\e i, \e j) = \varphi(A \e i, A \e j)$, e quindi le colonne di $A$ si mantengono a due a due ortogonali tra di loro, mentre $\varphi(A \e i, A \e i) = \varphi(\e i, \e i) = 1$. + Pertanto le colonne di $A$ formano una base ortonormale di $V$. \\ + + Si osserva che anche $A^\top \in U_n$. Allora le righe di $A$, che non sono altro che + le colonne di $A^\top$, formano anch'esse una base ortonormale di $V$. \\ + + \leftproof Nel moltiplicare $A^*$ con $A$ altro non si sta facendo che calcolare il prodotto + hermitiano $\varphi$ tra ogni riga coniugata di $A^*$ e ogni colonna di $A$, ossia $(A^* A)_{ij} = \varphi((A^\top)_i, A^j) = \varphi(A^i, A^j) = \delta_{ij}$. + Quindi $A^* A = A A^* = I_n$, da cui si deduce che $A \in U_n$. \end{proof} + \hr + \begin{note} + D'ora in poi, qualora non specificato diversamente, si assumerà che $V$ sia uno spazio + euclideo, reale o complesso. + \end{note} + \begin{definition} (norma) + Sia $(V, \varphi)$ un qualunque spazio euclideo. Si definisce \textbf{norma} la mappa + $\norm{\cdot} : V \to \RR^+$ tale che $\norm{\v} = \sqrt{\varphi(\v, \v)}$. + \end{definition} + + \begin{definition} (distanza tra due vettori) + Sia $(V, \varphi)$ un qualunque spazio euclideo. Si definisce \textbf{distanza} la mappa + $d : V \times V \to \RR^+$ tale che $d(\v, \w) = \norm{\v - \w}$. + \end{definition} + + \begin{remark}\nl + \li Si osserva che in effetti $\varphi(\v, \v) \in \RR^+$ $\forall \v \in V$. Infatti, sia + per il caso reale che per il caso complesso, $\varphi$ è definito positivo. \\ + \li Vale che $\norm{\v} = 0 \iff \v = \vec 0$. Infatti, se $\v = \vec 0$, chiaramente + $\varphi(\v, \v) = 0 \implies \norm{\v} = 0$; se invece $\norm{\v} = 0$, + $\varphi(\v, \v) = 0$, e quindi $\v = \vec 0$, dacché $V^\perp = \zerovecset$, essendo + $\varphi$ definito positivo. \\ + \li Inoltre, vale chiaramente che $\norm{\alpha \v} = \abs{\alpha} \norm{\v}$. + \end{remark} + + \begin{proposition} (disuguaglianza di Cauchy-Schwarz) + Vale che $\norm{\v} \norm{\w} \geq \abs{\varphi(\v, \w)}$, $\forall \v$, $\w \in V$, dove + l'uguaglianza è raggiunta soltanto se $\v$ e $\w$ sono linearmente dipendenti. + \end{proposition} + + \begin{proof} + Si consideri innanzitutto il caso $\KK = \RR$, e quindi il caso in cui $\varphi$ è + il prodotto scalare standard. Siano $\v$, $\w \in V$. + Si consideri la disuguaglianza $\norm{\v + t\w}^2 \geq 0$, valida + per ogni elemento di $V$. Allora $\norm{\v + t \w}^2 = \norm{\v}^2 + 2 \varphi(\v, \w) t + \norm{\w}^2 t^2 \geq 0$. L'ultima disuguaglianza è possibile se e solo se $\frac{\Delta}{4} \leq 0$, e quindi se e solo + se $\varphi(\v, \w)^2 - \norm{\v}^2 \norm{\w}^2 \leq 0 \iff \norm{\v} \norm{\w} \geq \varphi(\v, \w)$. + Vale in particolare l'equivalenza se e solo se $\norm{\v + t\w} = 0$, ossia se $\v + t\w = \vec 0$, da cui + la tesi. \\ + + Si consideri ora il caso $\KK = \CC$, e dunque il caso in cui $\varphi$ è il prodotto hermitiano + standard. Siano $\v$, $\w \in V$, e siano $\alpha$, $\beta \in \CC$. Si consideri allora + la disuguaglianza $\norm{\alpha \v + \beta \w}^2 \geq 0$, valida per ogni elemento di $V$. Allora + $\norm{\alpha \v + \beta \w}^2 = \norm{\alpha \v}^2 + \varphi(\alpha \v, \beta \w) + \varphi(\beta \w, \alpha \v) + \norm{\beta \w}^2 = \abs{\alpha}^2 \norm{\v}^2 + \conj{\alpha} \beta \, \varphi(\v, \w) + + \alpha \conj{\beta} \, \varphi(\w, \v) + \abs{\beta}^2 \norm{\w}^2 \geq 0$. Ponendo allora + $\alpha = \norm{\w}^2$ e $\beta = -\varphi(\w, \v) = \conj{-\varphi(\v, \w)}$, si deduce che: + + \[ \norm{\v}^2 \norm{\w}^4 - \norm{\w}^2 \abs{\varphi(\v, \w)} \geq 0. \] + + \vskip 0.05in + + Se $\w = \vec 0$, la disuguaglianza di Cauchy-Schwarz è già dimostrata. Altrimenti, è sufficiente + dividere per $\norm{\w}^2$ (dal momento che $\w \neq \vec 0 \iff \norm{\w} \neq 0$) per ottenere + la tesi. + \end{proof} + + \begin{proposition} (disuguaglianza triangolare) + + \end{proposition} \end{document} \ No newline at end of file