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--- a/euclidean-spaces.tex +++ b/euclidean-spaces.tex @@ -750,7 +750,7 @@ $$ Применяя к этому равенству отображение $\ph$ и пользуясь его линейностью, получаем \begin{align*} \ph(v) &= \ph(e_1 B(e_1,v) + e_2 B(e_2, v) + \dots + e_n B(e_n,v)) \\ -&= \ph(e_1)B(e_1,v) + \ph(e_2)B(e_2,v) + \dots + \ph(e_n B(e_n) \\ +&= \ph(e_1)B(e_1,v) + \ph(e_2)B(e_2,v) + \dots + \ph(e_n)B(e_n) \\ &= B(e_1\overline{\ph(e_1)} + e_2\overline{\ph(e_2)} + \dots + e_n\overline{\ph(e_n)},v). \end{align*} Заметим, что первый аргумент полученного выражения не зависит от $v$. @@ -830,7 +830,7 @@ $U\leq V$~--- конечномерное подпространство в $V$. \begin{enumerate} \item\label{num:orth-comp-prop-findim-1} $V = U\oplus U^\perp$; \item если, кроме того, $V$ конечномерно, то $\dim (U^\perp) = \dim (V) - \dim (U)$; -\item $(U\perp)^\perp = U$. +\item $(U^\perp)^\perp = U$. \end{enumerate} \end{proposition} \begin{proof} @@ -927,7 +927,7 @@ $U\leq V$~--- конечномерное подпространство, $v\in V $w_1+w_2\in U^\perp$. По определению $\pr_U(v_1) = u_1$, $\pr_U(v_2) = u_2$ и $\pr_U(v_1+v_2) = u_1 + u_2 = \pr_U(v_1) + \pr_U(v_2)$. -Мы показали аддитивность отображения $\pr_U$. Если $v\in U$ +Мы показали аддитивность отображения $\pr_U$. Если $v\in V$ и $v = u + w$ для $u\in U$, $w\in U^\perp$, то $v\lambda = u\lambda + w\lambda$, откуда следует и однородность $\pr_U$. @@ -1039,7 +1039,7 @@ $(\eta\circ\ph)^* = \ph^*\circ\eta^*$ \begin{align*} B(v,(\ph+\psi)^*(v')) &= B'((\ph+\psi)(v),v') \\ &= B'(\ph(v) + \psi(v),v') \\ -&= B'(\ph(v),v') + B(\psi(v),v') \\ +&= B'(\ph(v),v') + B'(\psi(v),v') \\ &= B(v,\ph^*(v')) + B(v,\psi^*(v')) \\ &= B(v,\ph^*(v')+\psi^*(v')), \end{align*}