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Se invece $h(x) = (x \cdot u_x)^2$, necessariamente $\norm{x}^2 = 1$, + e quindi $x \in S^{n-1}$. Ma allora: + \[ + x \cdot u_x = \frac{1 - x \cdot f(x)}{\norm{x - g(x)}}. + \] + Dal momento che $x$ non può essere un punto fisso di $f$, $x \cdot f(x)$ \underline{non} può essere uguale a $1$; quindi + $x \cdot u_x > 0$ per $x \in S^{n-1}$. Si conclude allora che $h(x) > 0$ per ogni $x \in D^n$, e quindi $f$ + è una funzione liscia. \smallskip + + Si verifica facilmente che $\restr{f}{S^{n-1}} = \id_{S^{n-1}}$. Questo tuttavia è un assurdo + per il Lemma \ref{lem:non_retrazione}, \Lightning. Quindi $f$ ammette almeno un punto fisso. \end{proof} \begin{theorem}[del punto fisso di Brouwer] @@ -960,7 +1010,32 @@ \end{theorem} \begin{proof} - ... + Supponiamo per assurdo che $f$ \underline{non} ammetta alcun punto fisso. Poiché $D^n$ + è compatto, per il Teorema di approssimazione di Weierstrass, per ogni $\eps > 0$ + esiste una funzione polinomiale $P_\eps : \RR^n \to \RR^n$ tale per cui + $\norm{P_\eps(x) - f(x)} < \eps$ per ogni $x \in D^n$. \smallskip + + Osserviamo che: + \[ + \norm{P_\eps(x)} \leq \norm{P_\eps(x) - f(x)} + \norm{f(x)} < \eps + 1, + \] + quindi $Q_\eps \defeq \frac{1}{\eps + 1} P_\eps$ si restringe su $D^n$ a + un'endofunzione $\restr{Q_\eps}{D^n} : D^n \to D^n$. \smallskip + + Dal momento che $f$ non ammette alcun punto fisso, $\mu \defeq \min_{x \in D^n} \norm{f(x) - x}$ è + tale per cui $\mu > 0$. \smallskip + + Osserviamo che: + \begin{equation*} + \begin{aligned} + \|Q_\eps(x) - f(x)\| & = \frac{1}{\eps + 1} (\|P_\eps(x) - f(x)\| + \eps \|f(x)\|) \\[0.1in] + & \leq \frac{2 \eps}{\eps + 1}. + \end{aligned} + \end{equation*} + Scegliendo allora $\eps$ tale per cui $\frac{2 \eps}{\eps + 1} < \mu$, $Q_\eps$ \underline{non} può ammettere + punti fissi: un punto fisso violerebbe infatti la minimalità di $\mu$ secondo la scorsa disuguaglianza. + Tuttavia $Q_\eps$ è una funzione liscia, e per il Lemma \ref{lem:punto_fisso_cinf} deve ammettere + punti fissi, \Lightning. Dunque $f$ ammette almeno un punto fisso. \end{proof} \section{Teoria del grado modulo \texorpdfstring{$2$}{2}} @@ -1000,7 +1075,7 @@ poiché i valori regolari sono densi, possiamo prendere $y'$ valore regolare di $H$. \smallskip Allora $H\inv(y')$ è una varietà di dimensione $1$, il cui bordo ha un numero pari - di punti per il Teorema \ref{thm:classificazione_dim_1}. \smallskip + di punti per il Corollario \ref{cor:classificazione_dim_1}. \smallskip Osserviamo che $\partial (M \times [0, 1]) = M \times \{0\} \sqcup M \times \{1\}$. Allora: \[