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Dann ist - \begin{align*} - \Vert (\mathbb{I} + B) x \Vert \qquad - &= \qquad \Vert x + B x\Vert \\ - &\stackrel{\text{Dreiecksungl.}}{\ge } \qquad \Vert x \Vert - \Vert Bx \Vert \\ + \begin{salign} + \Vert (\mathbb{I} + B) x \Vert + &= \Vert x + B x\Vert \\ + &\stackrel{\text{Dreiecksungl.}}{\ge } \Vert x \Vert - \Vert Bx \Vert \\ &\stackrel{\Vert Bx \Vert \le \Vert B \Vert \Vert x \Vert}{\ge } - \qquad \Vert x \Vert - \Vert B \Vert \cdot \Vert x \Vert \\ - &= \qquad ( \underbrace{1 - \Vert B \Vert}_{> 0}) \Vert x \Vert - .\end{align*} + \Vert x \Vert - \Vert B \Vert \cdot \Vert x \Vert \\ + &= ( \underbrace{1 - \Vert B \Vert}_{> 0}) \Vert x \Vert + .\end{salign} Also hat die Gleichung $(\mathbb{I} + B) x = 0$ nur die Lösung $x = 0$, also ist $(\mathbb{I} + B)$ injektiv und mit \ref{lemma:linabb} regulär. Bleibt zu zeigen: $\Vert (\mathbb{I} + B)^{-1} \Vert \le \frac{1}{1 - \Vert B \Vert}$. Es gilt - \begin{align*} - 1 \qquad &= \qquad \Vert \mathbb{I}\Vert \\ - &= \qquad \Vert (\mathbb{I} + B) (\mathbb{I} + B)^{-1} \Vert \\ - &= \qquad \Vert (\mathbb{I} + B)^{-1} + B (\mathbb{I} + B)^{-1} \Vert \\ - &\stackrel{\text{Dreicksungl.}}{\ge } \qquad \Vert (\mathbb{I} + B)^{-1} \Vert + \begin{salign} + 1 &= \Vert \mathbb{I}\Vert \\ + &= \Vert (\mathbb{I} + B) (\mathbb{I} + B)^{-1} \Vert \\ + &= \Vert (\mathbb{I} + B)^{-1} + B (\mathbb{I} + B)^{-1} \Vert \\ + &\stackrel{\text{Dreicksungl.}}{\ge } \Vert (\mathbb{I} + B)^{-1} \Vert - \Vert B (\mathbb{I} + B)^{-1} \Vert \\ - &\ge \qquad \Vert (\mathbb{I} + B)^{-1} \Vert - \Vert B \Vert \cdot \Vert (\mathbb{I} + B)^{-1} \Vert \\ - &= \qquad (1 - \Vert B \Vert) \Vert (\mathbb{I} + B)^{-1} \Vert - .\end{align*} + &\ge \Vert (\mathbb{I} + B)^{-1} \Vert - \Vert B \Vert \cdot \Vert (\mathbb{I} + B)^{-1} \Vert \\ + &= (1 - \Vert B \Vert) \Vert (\mathbb{I} + B)^{-1} \Vert + .\end{salign} Damit folgt die Behauptung. \end{proof} diff --git a/analysisII.pdf b/analysisII.pdf index bace607a85902a3399e56894f047827e7c63f6d9..d4716bc1914414438d0ac41fc6a45a4a1714e5a0 100644 GIT binary patch delta 6295 zcmai$Wl$T?x9v;O;1)DEG)M_90g6L$D{jG|xI>GixTH{AgS(_iad!$7N{hP}ic_@c z>-)bucjnF9nfqbwv-Zq6bH1#z&+qd&L)|&Uhj6rp=YQ^>3ksz(Wj*JFoA{9rlLXJz zWDKu_mHobp2uHHeg%An5hbEp|-PiUo3oG^gP^pvMyuS*3m7a5)xth6vu(I?&i=XYB z(VLm^`CA!z2bQ`|wJjS%&&shAyt`W8+g^XPb`f}u{3Ry-E}dGjI3ZIz&2aona`S>F znFed-FXMxeDe!j&y2n!ocorTLIO>fsRP6nqi<{xyHOcqp@ZqEF;o(OjK4|cT^X>VF z-!+2(#bFNziz413jQFX`SXMwCNixgh5l)>6D9q%fd{skH@qy4QmJ0ku!Vnw*Yay+9 z9H0Tm=rE79qM>Pz0a0B~qPpRBjClJ<;cb|<0vRd5s@Ik6J0cq{rS4TazCGAITEe8= zi(}npj_X-%YW;je)e$;@)|VRS3!v_c7zq_3DkC2Myp^EL463@uj50rd4aR;FJ%#=J zXp7{qis=7q!i*YOMKau92#l|8!h_Sb?-%X5DOA7LH4B+XemUIfMf?okL^n??3}1@v zri`(M0Nye)s?@>vSbN)EfDnHy^i=B9r0aYQ>l{B?T@uD}$4d_8Gs})~YtsA9YmHJn zXIM{x1XFEyMu`BwSUu8&d^7Vg8#&(8+)VD27w6-YKrC>izIw!2kUsnxIdf`mrO2lB zPVSIV$!_snvO@b}}31(+ulsPnHZ6 zGCiAZwIMm+4yzN!o0YKPu$Hz+6;^c*q&2q;ZyGmnIr)9-Qs8{@&9SSUx0ld`r*$s!>~?!ZaZw6I+&A8Mi z3ZpL*h^PmL7b2)@@sL+2+Q?gGf~{);XKP%*5`i<^1CxuJg`6!-(R8Y^IrK71#8emP ztQP82o>PRx@Os7C!7ZfV9*WUY`(RoPwv3j4=%{0KnJ;2`OGWCD$PvF3_9}pbM%0sI zIH4ibK5H}iT05nSne!#ni;>e29aeP6!}aA!Z+Z%6JJS9nlIk2Mt&dvF`W6}%iYRnd zGK9lf2`O6JpX0U2xAhEI@(xv~8DalbVS$TDN#&_~mHZAbT2}u6`?`T)FO#YaGw&od zhMB{Vkn}HAn(7ygCqkI z*76PWl!Gt|g1~I9{K3(F7;6q{O9r6%E>tUJ*p$^qPkz(xSFirSxhr@W@s1?X(^AU; zKsz_w3(cjJ1qap?$REsXfJ!{j#tc@|!QPloMSk*cD8vV|-b=NsEM*ti;T-PGAm9-X zmwjv{(=(w`;Po6Fcgugzf?g>i{5u3PS~mEnQhs!;4>#5=7qbkMr&ZhnHZvFeytn_j zqKdPHTisql2#k7fsZZ01%rnpHB-JrbLz=-lxEX$pCGAcPF6fc&X-n=gIqkZa3#nZc zjJX$QVL2d*<9jZG04^pk5}pyWz`sB;Pb*62MeS36BlW76ZDRO6GoOhBCHLKCKuo0@ zBBsSdoIV&9V}z9LArxepj5Db!iLOl&lTOEyYSR0OfAu+OFk$YF;0?Hc)gfHzT{Wr3 z`$Xt6|6aJN{Sq1m)Q3SFqYfoOl6G5}eeIT^)9N z;6&^zPUz1&>TRZxE~TWbx4dfXWxYV}T20$l7l@#3Z^9O70>vEXvLI3N>Xt`IK)}xY zsaJBP9jKAkf!V;j9xGocwy{bEIInShZ_bWWr<+l^DBgwS`Q=s`sxJ955r#}8e;YNB zboYLyMGr6ze5s6^Lm@T_Z&fVsa;JOH2>0p>r>%&-iV2SgRRANHZ3=L)Qea5@+?wYe4*T)^ywlT0)vK}w0W;cr_xSc!*i!>wa!B>l?mE_0yYdZV?bY+h+& zktk=H9S2O5GY^SW#QN)n;YHYW6sbnY$S?OCtf_$Tq2%P2&3i%c1Dd%fO%Vp^nTTT5 z-d7&X>@{O{1Cqq}O zCIVry@8$k4J^tFk(xGtQ&pamY1NIO0cT`{=L%)kkmau8R4lj>NbaVLS>XnUT>sF_} z!x_DDH+0;qJ)}_94;PC7Om1$ToD&%!;A#BknqI;j`wI7PkFFL@hCr|UKC$%@QA9y} z=_;#`{3mbS%KbkKy^}ClD)g}IVBf1tGzTW3l*&H&;rH3KpIGwIY_5!Kx5*JT@&0r! 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