From b56e701fde26f4dfe4b72a3660ec330ba28a97e2 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 9 Jun 2020 18:51:17 +0200 Subject: [PATCH] correct spelling in korollar environment --- ana1.tex | 4 ++-- ana12.tex | 8 ++++---- ana13.tex | 4 ++-- ana4.tex | 8 ++++---- ana5.tex | 4 ++-- ana6.tex | 4 ++-- ana8.tex | 4 ++-- analysisII.pdf | Bin 618557 -> 618587 bytes lecture.cls | 2 +- 9 files changed, 19 insertions(+), 19 deletions(-) diff --git a/ana1.tex b/ana1.tex index 6fa720d..4952382 100644 --- a/ana1.tex +++ b/ana1.tex @@ -333,7 +333,7 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^ .\end{align*} \end{proof} -\begin{korrolar}[Integration von Potenzreihen] +\begin{korollar}[Integration von Potenzreihen] Es sei $\sum_{n=0}^{\infty} a_n(x - x_0)^{n}$ eine reelle Potenzreihe mit Konvergenzradius $\rho > 0$. Dann konvergiert $\sum_{n=0}^{\infty} a_n (x - x_0)^{n}$ in jedem Intervall $[x_0 - r, x_0 + r]$ für $0 < r < \rho$ gleichmäßig und für $[a,b] \subset \;]x_0 - \rho, x_0 + \rho[$ gilt @@ -341,7 +341,7 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^ \int_{a}^{b} \sum_{n=0}^{\infty} a_n(x - x_0)^{n} dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1} \Big|_{a}^{b} .\] -\end{korrolar} +\end{korollar} \begin{proof} Nur die gleichmäßige Konvergenz für $| x - x_0| \le r$ ist zu beweisen: Für $|x - x_0| \le r$, $r < \rho$ gilt: diff --git a/ana12.tex b/ana12.tex index bc59b21..c4f8eea 100644 --- a/ana12.tex +++ b/ana12.tex @@ -67,9 +67,9 @@ Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $ Also ist $f$ differenzierbar und $Df(x) = \nabla^Tf(x)$. \end{enumerate} \end{proof} -\begin{korrolar} +\begin{korollar} stetig partiell differenzierbar $\implies$ (total) differenzierbar $\implies$ partiell differenzierbar. Die umgekehrten Implikationen gelten im Allgemeinen nicht. -\end{korrolar} +\end{korollar} \begin{lemma}[Richtungsableitung]\label{lemma:richtungsableitung} Sei $D\in \R^n$ offen, $f \colon D \to \R$ im Punkt $x\in D$ differenzierbar. Dann gilt $\forall v \in \R^n$ mit $\norm{v}_2 = 1$ existiert die Ableitung in Richtung $v$ (sog. \underline{Richtungsableitung}) \[\pdv{f}{v}(x) \coloneqq \lim\limits_{t\searrow 0} \frac{f(x + tv) - f(x)}{t}\] und \[\pdv{f}{v}(x) = (\nabla f(x), v)_2\] @@ -87,11 +87,11 @@ Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $ &= (\nabla f(x),v)_2 \end{salign*} \end{proof} -\begin{korrolar} +\begin{korollar} Sei $\nabla f(x) \neq 0$. Dann ist der Winkel $\theta$ zwischen zwei Vektoren $v\in \R^n$ und $\nabla f(x) \in \R^n$ definiert durch \[\cos(\theta) = \frac{(\nabla f(x), v)_2}{\norm{\nabla f(x)}_2\cdot \norm{v}_2}.\] Damit gilt für $\norm{v}_2 = 1$ \[\pdv{f}{v}(x) \oldstackrel{\text{Lemma } \ref{lemma:richtungsableitung}}{=} (\nabla f, v)_2 = \norm{\nabla f(x)}_2 \cdot \norm{v}_2 \cdot \cos(\theta) \oldstackrel{\norm{v}_2 =1}{=} \norm{\nabla f(x)}_2 \cdot \cos(\theta)\] $\pdv{f}{v}(x)$ wird maximal, wenn $\cos(\theta) = 1$, also wenn $v$ und $\nabla f(x)$ die gleiche Richtung haben: d.h. der Vektor $\nabla f(x)$ ist die Richtung des stärksten Anstiegs von $f$ im Punkt $x$. -\end{korrolar} +\end{korollar} \begin{bem} \begin{enumerate} \item Es gibt Funktionen, für welche alle Richtungsableitungen existieren, die aber dennoch nicht (total) differenzierbar sind. diff --git a/ana13.tex b/ana13.tex index 95d6259..86bdc38 100644 --- a/ana13.tex +++ b/ana13.tex @@ -77,7 +77,7 @@ $D \subset \mathbb{K}^{n}$ heißt \underline{konvex}, genau dann wenn: für alle $x,x' \in D$ und für alle $\lambda \in [0,1]$ gilt $\lambda \cdot x + (1-\lambda)x' \in D$. \\ Geometrisch: für zwei Punkte in $D$ liegt die Verbindungsstrecke der beiden Punkte stets ganz in $D$. \end{definition} -\begin{korrolar} +\begin{korollar} Seien $D \subset \R^{n}$ offen, $f: D \to \R^{n}$ stetig differenzierbar. Sei $x \in D$ und $\varepsilon > 0$ sodass $K_{\varepsilon}(x) \subset D$. Dann gilt: \begin{salign*} \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall y \in K_{\varepsilon} @@ -88,7 +88,7 @@ \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall x,y \in D \end{salign*} mit $M \coloneqq \sup_{z \in D} \norm{J_{f}(z)}_{2}$, das heißt $f$ ist auf $D$ Lipschitz-stetig. -\end{korrolar} +\end{korollar} \begin{proof} Aus Lemma \ref{lemma:dreieck-integrale} folgt: \begin{salign*} diff --git a/ana4.tex b/ana4.tex index e37373d..b932a45 100644 --- a/ana4.tex +++ b/ana4.tex @@ -132,9 +132,9 @@ Widerspruch zu $x \in S_{1}$, also $m>0$. Dann für $x \neq 0$ ist Vektor $\frac{x \ }{\norm{x}_{\infty}} \in S_{1}$ und $m \leq \frac{\norm{x} \ }{\norm{x}_{\infty}}$ (nach Definition von $m$) und $0 < m \cdot \norm{x}_{\infty} \leq \norm{x}, \ x \in \K^{n}$. \end{proof} -\begin{korrolar} +\begin{korollar} Auf $K^{n}$ sind alle Konvergenzen in irgendeiner Norm äquivalent zur Konvergenz in der $\ell_{\infty}$-Norm. (= komponentenweiser Konvergenz) -\end{korrolar} +\end{korollar} \begin{bem} Obiger Satz gilt nicht für unendlich dimensionale Räume (wie z.B. $C[a,b]$ oder $R[a,b]$). Die endliche Dimension von $K^{n}$ ist entscheidend. @@ -180,12 +180,12 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{enumerate} \end{proof} -\begin{korrolar} +\begin{korollar} \begin{enumerate}[1)] \item Endliche Schnitte und beliebige Vereinigung von offenen Mengen sind wieder offen. \item (Beobachtung) Durchschnitt von unendlich vielen offenen Mengen braucht nicht offen zu sein. Z.B. $$ \overset{\infty}{\underset{n=1}{\bigcap}} \left]-\frac{1}{n}, 1 + \frac{1}{n}\right[ = [0,1]$$ ist nicht offen, da $K_{\varepsilon}(0) \not\subset [0,1], \forall \varepsilon > 0$. \end{enumerate} -\end{korrolar} +\end{korollar} \begin{definition}[Abgeschlossene Menge] Eine Teilmenge $A \subset \K^{n}$ heißt abgeschlossen, wenn ihr Komplement $A^{c} \coloneqq \K^{n} \setminus A$ offen ist. diff --git a/ana5.tex b/ana5.tex index 13a08cc..717883f 100644 --- a/ana5.tex +++ b/ana5.tex @@ -302,10 +302,10 @@ In unendlich dimensionalen Banach-Räumen wie z.B.: $C[a,b]$ ist dies nicht möglich. \end{bem} -\begin{korrolar} +\begin{korollar} Jede abgeschlossene Teilmenge einer kompakten Menge in $\mathbb{K}^{n}$ ist ebenfalls kompakt. -\end{korrolar} +\end{korollar} \begin{proof} Sei $M \subset \mathbb{K}^{n}$ kompakt und $A \subset M$ abgeschlossen. Wegen diff --git a/ana6.tex b/ana6.tex index 8d3a81d..28d8f57 100644 --- a/ana6.tex +++ b/ana6.tex @@ -49,7 +49,7 @@ \end{align*} \end{proof} -\begin{korrolar} +\begin{korollar} \begin{enumerate}[a)] \item Ein Skalarprodukt $(\cdot,\cdot)$ auf $V$ über $\K$ erzeugt eine Norm durch $\norm{x} \coloneqq \sqrt{(x,x)}, \ x \in V$. Falls ein normierter Raum $\left(V, (\cdot,\cdot)\right)$ vollständig ist, so heißt das Paar $\left(V, (\cdot,\cdot)\right)$ \underline{Hilbert-Raum}. \item Das euklidische Skalarprodukt $(\cdot,\cdot)_2$ auf $\K^n$ @@ -58,7 +58,7 @@ $$\norm{x}_2 \coloneqq \sqrt{(x,x)_2} = \sqrt{\sum_{i=1}^n |x_i|^2}.$$ $\left(\K^n, (\cdot,\cdot)_2\right)$ ist ein Hilbert-Raum. \end{enumerate} -\end{korrolar} +\end{korollar} \begin{proof} Normeigenschaften Definitheit und Homogenität folgen aus \ref{def:definitheit}-\ref{def:linear}. Die Dreicksungleichung folgt aus der Schwarz-Ungleichung. diff --git a/ana8.tex b/ana8.tex index 4848566..4fc2305 100644 --- a/ana8.tex +++ b/ana8.tex @@ -37,11 +37,11 @@ Damit folgt die Behauptung. \end{proof} -\begin{korrolar} +\begin{korollar} Sei $A \in \mathbb{K}^{n \times n}$ regulär und $\tilde A \in \mathbb{K}^{n \times n}$ s.d. $\Vert A - \tilde A\Vert < \frac{1}{\Vert A^{-1} \Vert}$. Dann ist $\tilde A$ regulär. -\end{korrolar} +\end{korollar} \begin{proof} Es ist $\tilde A = \tilde A + A - A = (\tilde A - A) + A = A diff --git a/analysisII.pdf b/analysisII.pdf index 8937fc57fdff3168a6e0eb5c942134cf186d116d..d3dfd23fa03c461e5b028c00c9dd8c41b6347fc6 100644 GIT binary patch delta 35430 zcmV(?K-a&$;3eDOC9v%B0XdhEOam!@-CSFD+_n*Z*RP=Sv^owM;tr^B&S`2pb(|`$ zs`T{K$-~;BWZEk#-IbEmZU1{e03??nK`t$Ch3^)&N59I z$haX-|M=|5>7#EBQ^weWWQVDgYVr7x>czL;eD&zqfB)U_L;rB{=*ge`mwfSeH>e)* z_`Bn$r%%2;#!F1u^4b6U==9)U2Lz9KLLAFXBvQ!CD3iRrJb3;?o}A&`*GZl+qm!TA z$V>dE!L(+{D*4;NUk6k}l7gFmMDaY6l1<(EnN>8&UGik_xwINMGStdE#gAAQUVUZn0Ay_t{iK05AIj_~_kkQ4}`f9aW zuCen}LT8BdY8L3D)EVQl(X#Jzkw2-d!dw>@>*aNMRUM{O^Tkj0yXM1xGxy=TsNSxN z@>NmcSHzPJ8Dk796AkSw(6}BODpSNXBOh)2Je#7dIqr<43fm~~1FmZ8c~%dCcp+S- zy_rEKGD&I1p z(R4Y^ha7W=*Yr4!$=P!Z8zAB8108_+!tBz{sG_V2uL|p;Sgj7pqUdX2Y6;;-Faidn zNeRh}kpAaRJE8s3EH2Ay|M0X{Wc(jGtro@l!s_wztOWI&O>~2Qd6tW@WgIh|aSCK4k-{$M3WW$Z-=eS#FFsfqGO~tH$f%thn@# zje}nQbST7P3A-47@C18+Unh+{)aLX&A_1Byscs{Jp2Ly4w`oy%+o#qst?e@YaF`Mx znaIq9BQot0{$S$!yJwB;pi-oH|3fTDW;yh_9&r9WQ#?WiFhd#LH+~WZIo=NV9G*<+ zg^kX+Xg)dbGMt~OEa$DFUaX4prIp>Q%kLTIU`Le~9^T!5RA>HSj7bd-)o%AreOT7J z^CGfZhK=gWj0bis7!SPWkcJP^JtBQ>^oP+-%kqSB!k5AI?p-_Mf)Wmp_I9nWpzUs4 zl&eU{NGSz7a0OTr42$@&|BQGl&rlaSz3@R6o{nm~$TOlu=;nj41NQ~cLtSCrY9Cx5sMwT;J+O`F-VjkE5uL5-xBDME;mJ+w@M#Y1Bp z9Y5x1vSPE%tD(FB5cCOA$`MONP3p_y=c3$9Cggw|7B|-NI=%r;;kj9ym4$uKaS&3! zdukJZm^R1Kb≪bb(2rR4}@%tpL29iBx-lpYK#lY%;%{UItSQ6!W!R9mA}Y$7aQ1<{)DZ`Nm26#vhRaxqlu6Kh~L&sj)w#kk}5sGxpAvAx%|CZ9h+fw z>UU^#3TUsJL|a%RYXm~v!AgJwyGx}Rld``zEYL|oVtW&+C8d2L(j9kL0X8%cIgF7Z z3}m6yVM3ijJNf-QhJ91G55=Lp$mMl^i@jPT8O#Vr1FP%?#*fs*`Lv1e%m8*&!^AiW z890I0zU%;1A-f7%>&UDYyVHp6b!Ct$lPMcxsXa511LqTJm8G=RUl!Ywe0wg}aS4G| z!5EPTIHgjPN=0)38ojT|)`wK8p-xdqL9}3##>)C2MV(7xx$dmKR~wT&lhn3zV`jA% zHReiZiqVfjjRQS^Cw2t_;4of)d*uJdB!-Y{Q(I-D6KmSE0HdfRh#~}0q;AskA_}UK zKnP5>zjbr`=WS#XO*5U-+f4%d$N@BiN&Xj8X}B)Up(3!O=FH*qp-Nh4)JEAD*5ulG zKng!V!rYvVY4*KnfMSTj7G>ED&>huahWLj%D#st#`;4(u+^b=q2*8?uMFpwc9wuHC zSC_^0y4<`xWMttQbhKEUxL%!sufl#)W9hX&&v&`SqgXqH0!PdqTCq$G!2%!4RJLG2 zk4@JB3u3I`bQIw^II@i&`ywaW;_e4S#bVamvxK>Budz5wI5GsI63#h9Iq9H84k#r; z2NiZ?)T9`^aN`M9<>g_23UgX)ZhrJEnhxxz@QN@)*a_w10T5{cfx==T4dVMTvJg$& z2?oa=5(N2)h1Auq(9;HqeM76elAXE6a=1S3a=&g`Q$-Uftwk)Y2{lH*TU*m5<5lZG zYetLJUD?7L^KEP8oXq-@2y;-ytsLqz9M@=4l(8dXI3m4mzWT&}Gx$XmD%#3Q+cT32 zA=|JYo(^R)=Yh?+u_l()5X2!mff1}uik(LiDx)koGq0s9!iFuLo8DAC!I5nX?5AF4 zo9rx$m1-8Xey-Q40l#1e=5JW}%d{0aQ?h#`Og)Q8ckXDT*``n05`Q=eO+9Czr8-HFKQfWwz#@kmqqPp z|5zqN%sgK_bX29pG{9{))f?OCUsUPI z^5(MM;=D_i01|W6Mu?Ld=$T*)@`!$uT1W*3nbVQp!X7PyFc>rj&bp#PHQXuGaC&1f zl+M4R!URQs%Bl8yMWw@N_lk;??nHX_6%|WH8L{6xD(^4E@jEJv1L-wKh7Sw4NM4ox zmI=3gN{E1yT`9=wIxygoy&ez=MX=H>nmLc*YMZII*gR!iM0;ViYtY$4FDy{ocEG&| zjIep>wrlH8yKzLM)s$}wC%xV?169;K@$Xc@dW+oszcrPwl7ip|a4 z%f9ge{l)671N_Ztwi_7lh3$ckl;zAdL*rq8+Z6s~aapY2+*J0)j4>8+=xl^qPk3jP^6Tx~)6)rEx=ga`PhsQh>BJzD$W27m39MzSa`3U}-8dS$O6zO;|8nx72) z<8sx!Yb!B$_Lm1Q>kE+zK30qHKID(9;uqgQy1sVJx^1G-W&q)$Qi51#H4Av$p3C@u zGDvjg4{c*G6JWRSZNWvgE-s2y<-a|@zxEGj%Z-0_1s>;2z$%9^}j6lJV|I<%QI)t`n$h`PP-1vf9**x0|c# ztXS7F8)^A8%t7tcT@Ko(9f!#d<@N}F4M05yQ0J4E$+2(DOdUBjv}}TxH>W@!Ni-w< zbHYgf*xntgp7N(1$&8)_{HpQp-Q*v|_S(G5sbU$e;mPj5D3Fs~_n>bbhPc@`2Jm`m zZTxKHL^!7t^x!ac#^pL@QFcT2QoezOrja<`#7V9t@o2;!o5zX2aRBjk#x zxh@7QFqJ))Dvgif9uJv=-5^s%-RN1Wz*a`tv& z7?DCwjIzXPJ~_LXyqa>t$?;5UJv}>p`pvhe&z_tuI3k$%|udrr$q%^zih_i>HqtpTaO!uBxI5xiLgjQ4z+CA5-&UGLuvjBiLldEuq!s z|DIoDo9$1Vn;(z4o&M9WsHV3!$i59H)XHl159J6Mjj(@2x_SUdrf@_`Rt*Cw zo*H;sPF^1Uml+oVHGg9ZtsxdhMT2hE7npH@N7)D~O0$yn-d|HL5&kei3>Ux!EBHOx zY!f!8q9KJNO1T4wMm^I(M8n=qG*rrFKRyx-Z=2Wi(n`V)z;HWGz>LnABuwfs>QVNZ z3c1d&^Q(2R-oN-?z=!TxYeksS?ZZ}yVKDJogVisltVJiWUzg1n0z?aM!jj`5TGpcW z+3}Yg7y=mq)|WOI0waHS0AuXX6bg3Gs5LSLP+^Cu=%2y>Qgi!nq{2JZ|IL(u{j>nl z(~fYfhG!JCIFb=ldK0UigZ~Golpv9|5V&+sz^gD1HfKYo|9ka&AWSGW*<_QqyD0NO z*Uv(&9;OU9>{y4g?wtNi90=q6IaqiA_%)Ky#-r_#iG@3mMzhn9cg8+oY@O_(uVV()Y0E;d&|3qPW!0urF z!3h+eAlodm^=-Be9m*o3d1U2Rt3>9vsZ=bQYOW97kaR*Z$o zufw;u@cG97c9VZBJPW>xF%Eq7&WaJIO8XHJ6dX%Ij*wFb2{3~w;fQLR`VZ8WGmy;! z3dw%o3UF6WqA@eSMG`3RUiORdcv);<^^?cE)2o;>VcS{3)c3_6@@=*Wzry)lMNAz* zY)RqF22%|F4h&d{4<2T#Y;$?DUVscL2Ns@939xfW3>|;OcJPOiRnO1!MM0lc)0gw> z+wio=m)GHGotLlz{we=dMia&J%^GB5nb!dt{OKZJW$P_qjCmnv@Rt`}I&N0wC(p+B zdHLa%JS13P1`u=iP|_kGdu}d4tHrO!_h2 zzZC*nD%5{farU@0!U@TS`WG1Wy20&z)Z2qlpUmWCt(4|R#Sy?mbbwhlJGsX17^+@&#|fj66URo2ny8 z?`OPkr>Z9rDXrKe5t(LrIVhnKy2$cM?iyKO9S{T-*$1^DhbgJ{)*f(&xyF(Q5Lfa+ z-vi4P*^nm1wdrmS;5keijTw+q)jB_$@Re(wWJWy91;FAZ&fER)O<`)mJu?c25dDAs z3p~YOnmjEQIC#GEp`3yP{gXpN#V7~@zmrs@>qxBoD1i`9*%S#T0*C5bEmj8w?)+p73eXBm_IZT`UQfgA?Ef#BMwQ z$W!!L0x$$zc#VYdJW@B_F$8Ui0U>N`-}n9LxLRfemN;iV6*UsB>Y+HB8kOX|YBN

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