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Cam\-bridge University Press, 2006. +\bigskip +Автор благодарен Алексею Степанову за исправленные неточности и доказательство следствия~\ref{cor:isometry-extension}. + %%======================================================================== \section{Квадратичные формы: начало} @@ -381,13 +384,11 @@ $\la a,b\ra\cong\la c,d\ra$ для некоторого $d\in k$. Из срав $$ s_v(u)=u-2\frac{\ph(u,v)}{\ph(v,v)}v. $$ -%% \begin{AS} Простое вычисление показывает, что отражение является изометрией. \begin{lemma} Пусть $v_1,v_2\in V$ и $\ph(v_1)=\ph(v_2)\neq 0$. Тогда существует композиция отражений, переводящая $v_1$ в $v_2$. \end{lemma} -%% \end{AS} \begin{proof} Если $\ph(v_1-v_2)\neq 0$, то подойдет отражение относительно $v_1-v_2$: $s_{v_1-v_2}(v_1)=v_2$. Если $\ph(v_1+v_2)\neq 0$, то подойдет композиция отражения относительно $v_1+v_2$ ($s_{v_1+v_2}(v_1)=-v_2$) @@ -397,7 +398,6 @@ $$ что невозможно. \end{proof} -%% \begin{AS} \begin{corollary} Любая изометрия невырожденного пространства есть композиция отражений. \end{corollary} @@ -413,7 +413,6 @@ $$ тех же самых отражений, рассматриваемых уже как преобразований всего пространства $V$. Перенося $S$ в другую часть, получаем, что и $T$ является композицией отражений. \end{proof} -%% \end{AS} \begin{theorem}[Витта о сокращении] Если $q\perp\ph_1\cong q\perp\ph_2$, то $\ph_1\cong\ph_2$. @@ -434,8 +433,7 @@ $\psi_2(v_2)=a$ и $\psi_2(Tv_1)=a$. По лемме найдется изоме Это означает, что ограничение $S^{-1}T$ на $W_1$ и дает нужную изометрию между $\ph_1$ и $\ph_2$. \end{proof} -%% \begin{AS} -\begin{corollary}[{\bf Теорема о продолжении изометрии}] +\begin{corollary}[{\bf Теорема о продолжении изометрии}]\label{cor:isometry-extension} Пусть $(V,\ph)$~--- квадратичное пространство, $W_1,W_2$~--- подпространства в $V$ такие, что существует изометрия $\a\colon W_1\to W_2$. Тогда существует изометрия $\b\colon V\to V$ такая, что $\b|_{W_1}=\a$. @@ -460,7 +458,6 @@ $$ В этом случае нетрудно распространить изометрию на невырожденное подпространство, порожденное $u_1,\dots,u_{2s},v_1,\dots,v_r$, а затем использовать теорему о сокращении. \end{proof} -%% \end{AS} \begin{corollary} Любая невырожденная форма $\ph$ представляется в виде