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Из доказательства леммы следует, что найдется +Возьмем $v\in V$ такой, что $\ph(Tv)=\ph(v)\neq 0$. По лемме найдется композиция отражений $S\colon V\to V$ такая, что $Sv=Tv$. Отображение $S^{-1}T$, таким образом, является изометрией и оставляет $v$ на месте; значит, $S^{-1}T$ оставляет на месте и $W=(kv)^\perp$~--- подпространство размерности $n-1$. По предположению индукции изометрия $S^{-1}T|_W$ является композицией @@ -409,6 +415,8 @@ $$ тех же самых отражений, рассматриваемых уже как преобразований всего пространства $V$. Перенося $S$ в другую часть, получаем, что и $T$ является композицией отражений. \end{proof} +%% \end{AS} + \begin{theorem}[Витта о сокращении] Если $q\perp\ph_1\cong q\perp\ph_2$, то $\ph_1\cong\ph_2$. \end{theorem} @@ -427,14 +435,35 @@ $\psi_2(v_2)=a$ и $\psi_2(Tv_1)=a$. По лемме найдется изоме кроме того, $S^{-1}Tv_1=v_2$, поэтому $S^{-1}T$ переводит $W_1=(kv_1)^\perp$ в $W_2=(kv_2)^\perp$. Это означает, что ограничение $S^{-1}T$ на $W_1$ и дает нужную изометрию между $\ph_1$ и $\ph_2$. \end{proof} + +%% \begin{AS} \begin{corollary}[{\bf Теорема о продолжении изометрии}] -Пусть $(V,\ph)$~--- квадратичное пространство, $W_1,W_2$~--- два изометричных подпространства -(то есть $\ph|_{W_1}\cong\ph|_{W_2}$. Тогда существует изометрия $\a\colon V\to V$ такая, -что $\a(W_1)=\a(W_2)$. +Пусть $(V,\ph)$~--- квадратичное пространство, $W_1,W_2$~--- подпространства в $V$ такие, +что существует изометрия $\a\colon W_1\to W_2$. Тогда существует изометрия $\b\colon V\to V$ такая, +что $\b|_{W_1}=\a$. \end{corollary} + \begin{proof} -Это просто другая переформулировка теоремы Витта о сокращении. +Как и в доказательстве теоремы Витта о сокращении можно считать, что форма невырождена +на $V$. +В случае, когда $W_1$ невырождено, утверждение следует из теоремы Витта о сокращении. +Действительно, в этом случае $V$ раскладывается в прямую сумму $W_i$ и его ортогонального +дополнения ($i=1,2$). По теореме о сокращении существует изометрия +$\gamma:W_1^\perp\to W_2^\perp$. Тогда $\b=(\a,\gamma)$~--- изометрия $V\to V$. + +Если же $W_1$~--- вырожденное подпространство в невырожденном пространстве +$V$, то можно выбрать базис $u_1,\dots,u_{2m},v_1,\dots,v_k$ пространства $V$, содержащий базис +$u_1,u_3,\dots,u_{2s-1}$,\allowbreak $v_1,\dots,v_r$ пространства $W_1$ такой, +что матрица формы $\ph$ в этом базисе будет иметь вид +$$ +\operatorname{diag}\left(\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right),\dots, +\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right),\alpha_1,\dots,\alpha_k\right). +$$ +В этом случае нетрудно распространить изометрию на невырожденное подпространство, порожденное +$u_1,\dots,u_{2s},v_1,\dots,v_r$, а затем использовать теорему о сокращении. \end{proof} +%% \end{AS} + \begin{corollary} Любая невырожденная форма $\ph$ представляется в виде $$\ph\cong\underbrace{\mathbb H\perp\dots\perp\mathbb H}_{r\text{ раз}}\perp\ph_{an},$$