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Altrimenti, è sufficiente dividere per $\norm{\w}^2$ (dal momento che $\w \neq \vec 0 \iff \norm{\w} \neq 0$) per ottenere - la tesi. + la tesi. Come prima, is osserva che l'uguaglianza si ottiene se e solo se $\v$ e $\w$ sono + linearmente dipendenti. \end{proof} \begin{proposition} (disuguaglianza triangolare) - + $\norm{\v + \w} \leq \norm{\v} + \norm{\w}$. \end{proposition} + + \begin{proof} + Si osserva che $\norm{\v + \w}^2 = \norm{\v}^2 + \varphi(\v, \w) + \varphi(\w, \v) + \norm{\w}^2$. + Se $\varphi$ è il prodotto scalare standard, si ricava che: + \[ \norm{\v + \w}^2 = \norm{\v}^2 + 2 \varphi(\v, \w) + \norm{\w}^2 + \leq \norm{\v}^2 + 2 \norm{\v} \norm{\w} + \norm{\w}^2 = + (\norm{\v} + \norm{\w})^2,\] + + dove si è utilizzata la disuguaglianza di Cauchy-Schwarz. Da quest'ultima disuguaglianza si ricava, prendendo la radice quadrata, la disuguaglianza + desiderata. \\ + + Se invece $\varphi$ è il prodotto hermitiano standard, $\norm{\v + \w}^2 = \norm{\v}^2 + 2 \, \Re(\varphi(\v, \w)) + \norm{\w}^2 \leq \norm{\v}^2 + 2 \abs{\varphi(\v, \w)} + \norm{\w}^2$. Allora, riapplicando + la disuguaglianza di Cauchy-Schwarz, si ottiene che: + + \[ \norm{\v + \w}^2 \leq (\norm{\v} + \norm{\w})^2, \] + + da cui, come prima, si ottiene la disuguaglianza desiderata. + \end{proof} + + \begin{remark} + Utilizzando il concetto di norma euclidea, si possono ricavare due teoremi fondamentali della geometria, + e già noti dalla geometria euclidea. \\ + + \li Se $\v \perp \w$, allora $\norm{\v + \w}^2 = \norm{\v}^2 + \overbrace{(\varphi(\v, \w) + \varphi(\w, \v))}^{=\,0} + \norm{\w}^2 = \norm{\v}^2 + \norm{\w}^2$ (teorema di Pitagora), \\ + \li Se $\norm{\v} = \norm{\w}$ e $\varphi$ è un prodotto scalare, allora $\varphi(\v + \w, \v - \w) = \norm{\v}^2 - \varphi(\v, \w) + \varphi(\w, \v) - \norm{\w}^2 = \norm{\v}^2 - \norm{\w}^2 = 0$, e quindi + $\v + \w \perp \v - \w$ (le diagonali di un rombo sono ortogonali tra loro). + \end{remark} + + \begin{remark} + Sia $\basis = \{ \vv 1, \ldots, \vv n \}$ è una base ortogonale di $V$ per $\varphi$. \\ + + \li Se $\v = a_1 \vv 1 + \ldots + a_n \vv n$, con $a_1$, ..., $a_n \in \KK$, si osserva + che $\varphi(\v, \vv i) = a_i \varphi(\vv i, \vv i)$. Quindi $\v = \sum_{i=1}^n \frac{\varphi(\v, \vv i)}{\varphi(\vv i, \vv i)} \, \vv i$. In particolare, $\frac{\varphi(\v, \vv i)}{\varphi(\vv i, \vv i)}$ è + detto \textbf{coefficiente di Fourier} di $\v$ rispetto a $\vv i$. Se $\basis$ è ortonormale, + $\v = \sum_{i=1}^n \varphi(\v, \vv i) \, \vv i$. \\ + + \li Quindi $\norm{\v}^2 = \varphi(\v, \v) = \sum_{i=1}^n \frac{\varphi(\v, \vv i)^2}{\varphi(\vv i, \vv i)}$. In + particolare, se $\basis$ è ortonormale, $\norm{\v}^2 = \sum_{i=1}^n \varphi(\v, \vv i)^2$. In tal caso, + si può esprimere la disuguaglianza di Bessel: $\norm{\v}^2 \geq \sum_{i=1}^k \varphi(\v, \vv i)^2$ per $k \leq n$. + \end{remark} \end{document} \ No newline at end of file

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