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z=0YQ_QtbQ|_a!0oB?zXMyTahR3beZhZ1Q5V1#!X#EUEi?6Si^hNGyG)fZmHvh~wO4 z&EKbNUe$9`v#GS51OAkw6?D07+c z-;pj#0)(QyU+3HUJwWfG2+$b(l130V1vf77pTeBU0OwO-wV*p>$bGL!XUB3rBy%X~ zoDzVXN}kMU;8o$W7&`=HSF+2K*j=4UX;uG!_W?2C>id{e3;8K*odM1MCyLVjk^ zHnI;n{jYu2A3A@9e18=7_xM>U4ooGFX5jPR1aqrUTb%C8=KQrL#Ib~B3UfH4pGQUY zKhh;!t8%9V)+ZMRGctKpFf3S$b&axVqyJJyP6^yif`p*+s(iGN-j|KY9_qm?i(uz; zJam&SVId~URd{(|I%!DGC-BP0dCDh&Hbf02g&FwQn&HAl4}hopht$DRrSX-x&;XJx9evcg9%Yi diff --git a/euclidean-spaces.tex b/euclidean-spaces.tex index 7769807..1956582 100644 --- a/euclidean-spaces.tex +++ b/euclidean-spaces.tex @@ -158,10 +158,10 @@ $B(iv,iv) = -B(v,v)$ для всех $v\in V$. $B(u,v) = \ol{u}^Tv$. Нетрудно видеть, что эта форма полуторалинейная \begin{align*} -&B(u,v_1+v_2) = u^T(v_1+v_2) = \ol{u}^Tv_1 + \ol{u}^Tv_2 = B(u,v_1) + +&B(u,v_1+v_2) = \ol{u}^T(v_1+v_2) = \ol{u}^Tv_1 + \ol{u}^Tv_2 = B(u,v_1) + B(u,v_2)\\ &B(u,v\lambda)=\ol{u}^T(v\lambda)=\lambda(\ol{u}^Tv)=\lambda B(u,v)\\ -&B(u_1+u_2,v) = \ol{(u_1+u_2)}^Tv = \ol{u_1}^tv + \ol{u_2}^Tv = B(u_1,v) +&B(u_1+u_2,v) = \ol{(u_1+u_2)}^Tv = \ol{u_1}^Tv + \ol{u_2}^Tv = B(u_1,v) + B(u_2,v)\\ &B(u\lambda,v)=\ol{(u\lambda)}^Tv=\ol\lambda(\ol{u}^Tv)=\ol\lambda B(u,v)\\ \end{align*} @@ -227,7 +227,7 @@ $||v|| = \sqrt{B(v,v)}$ \dfn{длиной}\index{длина вектора} $v$. \begin{lemma}\label{lem:triangle_inequality} Пусть $(V,B)$~--- эвклидово или унитарное пространство, $u,v,\in V$. Тогда \begin{enumerate} -\item ({\it Однородность нормы}). $||\lambda v|| = |\lambda|\cdot +\item ({\it Однородность нормы}). $||v\lambda|| = |\lambda|\cdot ||v||$ для любого $\lambda\in k$. \item ({\it Теорема Пифагора}). Если $B(u,v)=0$, то $||u+v||^2 = ||u||^2 + ||v||^2$. @@ -243,7 +243,7 @@ $|B(u,v)|\leq ||u||\cdot ||v||$, причем равенство достига Однородность нормы следует из полуторалинейности: $$ -||\lambda v||^2 = B(\lambda v,\lambda v) = +||v\lambda||^2 = B(v\lambda, v\lambda ) = \lambda\ol{\lambda}B(v,v) = |\lambda|^2\cdot ||v||^2. $$ @@ -253,15 +253,15 @@ $$ Для доказательства неравенства Коши--Буняковского--Шварца положим $$ -w = u - \frac{B(u,v)}{B(v,v)}v +w = u - v\frac{B(u,v)}{B(v,v)} $$ -и заметим, что $$B(w,v) = B(u-\frac{B(u,v)}{B(v,v)}v,v) +и заметим, что $$B(w,v) = B(u-v\frac{B(u,v)}{B(v,v)},v) = B(u,v) - \frac{B(u,v)}{B(v,v)}B(v,v) = 0.$$ Это означает, что векторы $v$ и $w$ ортогональны. Поэтому и вектор -$\frac{B(u,v)}{B(v,v)}v$ ортогонален вектору $w$. Применим к этой паре +$v\frac{B(u,v)}{B(v,v)}$ ортогонален вектору $w$. Применим к этой паре векторов теорему Пифагора: $$ -||u||^2 = ||w||^2 + ||\frac{B(u,v)}{B(v,v)}v||^2 = ||w||^2 + +||u||^2 = ||w||^2 + ||v\frac{B(u,v)}{B(v,v)}||^2 = ||w||^2 + \frac{|B(u,v)|^2}{||v||^2} \geq \frac{|B(u,v)|^2}{||v||^2}, $$ откуда $|B(u,v)|\leq ||u||\cdot ||v||$.