From 38f5896316c7a36cd4b443bf3c9944138f750754 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 3 Jun 2020 10:41:31 +0200 Subject: [PATCH] fix some wrong quotation marks --- ana1.tex | 10 +++++----- ana2.tex | 2 +- analysisII.pdf | Bin 602990 -> 602770 bytes 3 files changed, 6 insertions(+), 6 deletions(-) diff --git a/ana1.tex b/ana1.tex index da13f27..6fa720d 100644 --- a/ana1.tex +++ b/ana1.tex @@ -130,7 +130,7 @@ \addplot[domain=0:1.1,samples=50,smooth,dashed, blue] {-0.3}; \end{axis} \end{tikzpicture} - \caption{Links: ,,$\epsilon$-Schlauch'', Rechts: \ref{bsp:punktweisekonvergenz} (b) nicht gleichmäßig konvergent} + \caption{Links: ,,$\epsilon$-Schlauch``, Rechts: \ref{bsp:punktweisekonvergenz} (b) nicht gleichmäßig konvergent} \end{figure} Beispiele \ref{bsp:punktweisekonvergenz} (a) und (b) nicht gleichmäßig konvergent, zu (a): Für $\epsilon = \frac{1}{2}$ gilt @@ -206,12 +206,12 @@ \begin{proof} \begin{enumerate}[(i)] - \item ,,$\implies$'': Sei $\epsilon > 0$. Dann $\exists N \in \N$ s.d. + \item ,,$\implies$``: Sei $\epsilon > 0$. Dann $\exists N \in \N$ s.d. $|f_n(x) - f(x)| < \frac{\epsilon}{2}$ $\forall x \in [a,b]$ und $\forall n \ge N$. Damit folgt: \[ \max_{x \in [a,b]} |f_n(x) - f(x)| = \Vert f_n - f \Vert_{\infty} < \frac{\epsilon}{2} < \epsilon \implies \Vert f_n - f \Vert_{\infty} \xrightarrow{n \to \infty} 0 .\] - ,,$\impliedby$'': Sei $\epsilon > 0$. Wegen + ,,$\impliedby$``: Sei $\epsilon > 0$. Wegen $\Vert f_n -f \Vert_{\infty} \xrightarrow{n \to \infty} 0$ $\exists N \in \N$ s.d. $\Vert f_n - f \Vert_\infty < \epsilon$ $\forall n \ge N$. Damit folgt $\forall n \ge N$ und $\forall x \in [a,b]$ @@ -219,7 +219,7 @@ |f_n(x) - f(x)| \le \max_{x \in [a,b]} |f_n(x) - f(x)| = \Vert f_n - f \Vert_{\infty} < \epsilon \implies \text{gleichmäßige Konvergenz} .\] - \item ,,$\implies$'' $(f_n)_{n\in\N}$ konvergiert gleichmäßig auf $[a,b]$, d.h. + \item ,,$\implies$`` $(f_n)_{n\in\N}$ konvergiert gleichmäßig auf $[a,b]$, d.h. $\exists f: [a,b] \to \R$ s.d. $f_n \xrightarrow[\text{gleichmäßig}]{n \to \infty} f$. Sei $\epsilon > 0$, dann $\exists N_0 \in \N$ s.d. $|f_n(x) - f(x)| < \frac{\epsilon}{4}$ @@ -234,7 +234,7 @@ \le \frac{\epsilon}{2} < \epsilon \quad \forall n, m \ge N_0 .\end{align*} - ,,$\impliedby$'': Sei $\epsilon > 0$, dann $\exists N_0 \in \N$, s.d. + ,,$\impliedby$``: Sei $\epsilon > 0$, dann $\exists N_0 \in \N$, s.d. \[ |f_n(x) - f_m(x)| \le \Vert f_n - f_m \Vert_\infty < \frac{\epsilon}{2} \quad \forall n,m \ge N_0 \quad \forall x \in [a,b] diff --git a/ana2.tex b/ana2.tex index 866edc0..bbd124c 100644 --- a/ana2.tex +++ b/ana2.tex @@ -57,7 +57,7 @@ Jetzt: Fourier Analysis! \begin{definition} Wir definieren \[ - (f, g) := \int_{a}^{b} f(x) \overline{g(x)} \d x \qquad (\text{,,Sesquilinearform''}) + (f, g) := \int_{a}^{b} f(x) \overline{g(x)} \d x \qquad (\text{,,Sesquilinearform``}) .\] Dies ist wohldefiniert da für $f, g \in R[a,b]$ das Produkt $f(x) \cdot \overline{g(x)} \in R[a,b]$ ist. \end{definition} diff --git a/analysisII.pdf b/analysisII.pdf index 5d28e2374dfd0f309a911bae0f479e8df264cd6e..6949ead734065017869ac9f30972ca09fc940a8b 100644 GIT binary patch delta 21278 zcmYiN1yoeu_dgChz@?=@xzNJ*C<;Q#vm z{Jzh6*5b1EKKr$MpL2#`=J+Rl;tqWc%>eANo_1go6)AfU+XCd?LL#U~oy0r6lxZx0deTNCVJkaT=G(_C&v98OADj%E}<(y2}-1T{5 z+tU)vV}RAFqhrNYlcN4VZLw!=7a5$?PJZj(fc(Lyv_uV2zn`}qn{xyjT`imZ1YVpYMU{T;?3E{ruDOmZ z#ZdxyG2o!Iy8oreD;$B#K!38aFh=1TNvnxsQR6qS+rmmps@U9$&Nh)TQOZl{F4k+t zw+9)D$@s!V|6+tFcf0n<<_CRUuhmtvce^G&+->ap!y1C27ueh!3`lkVz-jU^((CF4 zEO=a-VdOelYV7Y4^gb|vFInfjyjB-I_LSy7!Qm)Kh|^Sj@-;Rltber7%*se~tlxLp zm9{pE-Icjdc46K)XQ7}J7QiHEJhJe%6-n*R&EhoB44miH{BX=#dJ7f`tq>YftlDZ+ zK8x!P^0%;5dBusyFgVc&kdbIy`u&m~@=Yj3+N@iaJwa(>pUD`Ytk(wsvm5ay-t|b*LtzJ?S+t8f|ABnt#K5DuOk8<`mxctIK*OKyKoqY1d0m zSDkye<3CUQML)SB^{xaWB5yXkf9Ug)(FuX)_sBI*UzQqcT&ym(BTo0IhUV&XTvyD7rc%%FozK{h6PClycpAx^YY|*e5xV~n7dSK$QI>KEPCly1n^|9S5BKBc<+4fD5lbL^{j%==3BY4--ds&0Fdkj*c67o8QIf8jI1 zSLNItkEJCFPJ^QXSAPW;!%@W!_BqlfG3ms+EY$qAP%jX;+$gdHSq$V+5p0TCXot1N zhiKl>MEM%Bz-_i$vr_d^Q)9NM*nbi-T3J!4kHY&&5P5hd!WpiXDAns!7)b*L9tNunGw1qaZ9~WR<7~zInJ0ep=!G<} zKYxvtGlQV=gmgo5Ni7`NnN-iZ+!qCdklFjD`bF}W-tlS@f}G7)Ds%@od!;dFAJS&c z#=M-KdYGqM$h?3BCmi$?h+z$XmUkn53Jea1eLOLI+UZ)GKc6B-%wIsn54SG0{XPC- zFMD;u&pC}`KhErFOtWzrm^C4%;LZ9ny5=Bp%-=?*-JaMN%TXs;oX`;!9ryJC^HMzK zxx~pgc^Nw=$7s10!aB~X+|mpw3&K2oGyQKtwSN|pNAJ%9yb}!0Yb|JuFpUMu6&nTl z8Az&@c-elLB-75Sb4nom^Z*7;)HonCmqEh#)pSSIS28lB){Z4Ud! z{zB=sQaH}3cFRqv$fA#0gvui2drg@WjalAxs|1;5g8+&{Lh%jHXO`#M)whJq6QUWp zLX7XFny%pwbdK!lWt~-v#Lk`z)2+6ABqe=w7OdUJnY0w~P6JOC1VtSJ+Bi>S_!4?^ zzOCp9${tD(&O9vho1cAM=a>2`9^&@F=)smRrGSoXS5U9qj)yX4U*~Y+@L{mqZ8C$U zXu1Fg=S$2_x|@$iZo_Sp^PiPf|7>$vC#fE8{I0ZN-2HuId_?A)GtEEu7lt)V59SF0 z_+gG%Fz7*m9z^ItYIei|@{zz(W+XO{lK^l9Pw@eK(3cKi2R8_SCNPHpAOy<^fdddE zgs3ti;1~Fb2*7Xt{0zW`fho@*^cXSl8*CwlD9t#44MZRX9>G2m=*HKeKw zGJv32kQ|VPfpp}Max?`1gMmI25GFAN!i-Y_!(a?0#FkP4Kfoj^2%bX?h{3>UYRK#f z4bTCu&_LoZv_L6XP76`*=^&sr9VB(}LSk>urUw8RXhIM1O$@*oSik^L6Z1%H&E1Ru z3Jjd>K>82K){OidK!$-sOpps|W`G?AVlqQeQx;$cRAqq-2h>AKA6WrK7%0yQ#k#=; zki)Btzy^;5 zfl)9=5Lz1(0xrR3A&7eK3$Xr^qJbzcfNAiHFtj#~3at@|Kt<~1V`G=dj1k<1_@q5km)Z_43yfy4Y>Xi z>I<_DzzhQkbs&kFALJq76)*{Ey@FJji6LN)F67@-7h>aY@yHoy*oXB$BwV>&J0F~K%Q-Ak<#_wPDOVG2tgGts0HS3z${4T_J6UsyF(Mt>kdi(8#;dvr~u*~ z{|8R;guptU5cn<}O18)gs0817K`FiU1{7f6OK&L1FYllnD&9dDXtG`18n$^*!5?xp&5FbUx(5K=Aa?*HP=Et}L0&jS zy$*zWFCPd|qCvnd_&f-r{@W@Z!O)EI2SaSBI%MAX15g4|e1KReTUJme1ZpZ$2oxsQ zM}QCpQhfYpK@%F6kWc^>2Fis(Awunk>Jk+OiT<0jop5M0i^CxnO5?vaBZGGlKq%M{ z0VVn0S@I1sQ0Kf~5Z$5R8Eg{EmjalEeb^Fc3KwvIR{u8|WVg zomOIT5c}WEwZ;R>pl>|HN=!oyoJ;sOD9H&B|8H1JXp$han+OQ_AYU?&3-+u*qZ@w& z4FdirKo$nhCqu#8rvTkBP%i~?CY}ng!9dnjC=CBJpdPeKgBE4dfkPO`l@5_onSdz* z$esylHD&|9U|@dsKQb5Cg#FvN(9Mz$G(tB^KD6pq2oyl4dLcwI7Xu~GHY^69?OXyB zLEE_mGM4oj&_VzsKSRzs-Jp=zD*y@@cw7$g#+ASks9XtwjH`ei7^qkU5Q0_z&N|Z? 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