彐a>0 such that (,0)a|lgg),Vv∈X 22 dx (Recall this Poincare-Friedricks inequality follows from the fact that v E X satisfy v(0)=v(1)=0. For our problem,a=2works, "as can be shown (say) by considering the Rayleigh quotient. 彐B(=1)>0 such that a(u,y)≤B‖ln(g)‖l(g) (Recall this is derived from the Cauchy-Schuvarz inequality The error e=u-uh satisfies ela()≤(x< degradation error in HI projection of u on Xh n general ah is not the H projection of u on Xh b Exercise 4 For what particular problem(give the strong form)is uh H projection of u on Xn?L Note 5 Proof of H norm general bound(Optional) To begin, we note that for any wn E Xh a( (u )(orthogonality Bla-wn H()lah -wnH(Q2)(continuity)❪✩❫✍❴✁❵❜❛✁❝●❞✍❝●❡❣❢✼❫✐❤❦❥♠❧♦♥✏♣✁♥rq▼s t✈✉❳✇✖①③②✐④◆⑤✙⑥✼⑦❜⑥✚⑧❇⑦ ❥♠❧⑩⑨✵♣✐⑨❶q❸❷ ✉❏❹ ⑨ ❹✆❺❻❨❼❩❽✞❾✚❿ ♣➁➀➂⑨➄➃✦➅ ➆✵➇➉➈ ➊ ⑨ ❺➋❚➌❱➍ ❷ ✉ ➆✚➇➎➈ ➊ ⑨ ❺➋❨➌➏➍➑➐ ➇➎➈ ➊ ⑨ ❺ ➌❱➍◆➒➓➒→➔ ➣⑩↔❴✙❛✙↕❇➙●➙➛❡●➜❶❝✷➝➟➞❨❫➠❝●➡✵❛✙↕❇❵➤➢❴✁➥✒➦✸❵▼❝✹❴❩➧➠❵▼❝✹❛❩➨❇➝➩❝●➡✵❴✙➫✆➭◆↕➠➙➯❝●❡❣❢➲❤✆❫➠➙●➙✞❫❇➳➤➝➵❤✙❵❜❫❇➸➺❡⑩➜✧❴③❤✆↕❱❛✆❡➻❡●➜◆↕❇❡➼⑨➽➃➾➅ ➝✁↕❇❡❣❝✷➝⑩❤✙❢✼⑨♠❧ ① q➑➚➪⑨♠❧♦➶➹q➑➚ ①✵➘ ➦✫❫➠❵➲❫❇➭❶❵➂➴✵❵❜❫❱➷✆➙✏❴✆➸➮➬ ✉ ➚ ➈❺✃➱➳❐❫➠❵❜➨➹➝✙➬❩❒✭↕❇➝➲❛✙↕❇➡❮➷✙❴✼➝❜➜✧❫❇➳✢➡ ➣➝✁↕❇❢✙❰➻➷✆❢➻❛✙❫➠➡✧➝▼❝✹➧➏❴✁❵▼❝●➡❅Ï➲❡⑩➜✧❴ ↔↕➠❢❇➙✞❴✆❝ÐÏ➹➜Ñ➫✆➭◆❫➠❡✒❝✹❴✁➡◆❡ ➘ ❰ ❪✩❫❇➡✚❡✒❝●➡✚➭❶❝●❡✒❢➻❫✐❤✭❥♠❧♦♥✏♣✁♥rq▼s t➓Ò ❧Ó➚Ô➶➹q ✇➎①③②❜④◆⑤✙⑥✼⑦❜⑥◆⑧➠⑦ ❥✵❧✹Õ➂♣✐⑨❶q▲Ö Ò❦❹ Õ ❹ ❻❼ ❽✞❾✚❿ ❹ ⑨ ❹ ❻❼ ❽✞❾◆❿ ➔ ➣⑩↔❴✙❛✙↕❇➙●➙✸❡⑩➜✔❝✷➝➼❝✷➝×➧➏❴✁❵▼❝●❞❇❴❩➧➓❤✙❵❜❫➠➸Ø❡⑩➜✧❴➄❪✩↕❇➭◆❛✙➜✔❢❇➥ÓÙ♠❛✙➜✔➳❐↕➠❵❜Ú➑❝●➡✵❴✙➫✁➭◆↕❇➙➯❝●❡❣❢➘ ❰ Û✵Ü✷Ý✸Ü✷Þ ß➲à✔á❂à❅â➹ã◆ä✄å✼à❅æ☞ç❂ä⑩è é✵ê❅ë●ì✫í❏î✔ï ð⑥✧ñ➼ñ✍ò❜ò❩ó➏ò▲ô ➚❮õ➄ö❏õ♠÷ ②❩⑧❇⑦❩ø✏②✐ù◆ñ✍② ❹✆ô❶❹ ❻❨❼❩❽✞❾✚❿ Ö ❧✐➶ ➐ Ò ✉ q ú û▼ü ý þ➹ÿ✁￾✄✂✆☎✙þ✝☎✟✞✡✠☞☛✄✌ ø✎✍✑✏ ✒✔✓✖✕✘✗ ❹ õ➄ö❏Õ❨÷ ❹ ❻❨❼▼❽✞❾◆❿ ú û▼ü ý ÿ✁✂✡✂✡☛✄✂✙✠☞✌✛✚✢✜✔✣✝✂✡☛✥✤✹ÿ✁✦✧✞✡✠☞☛✄✌★☛✥✩✫✪✛☛✄✌✛✬✮✭ ✯ ø✎✍✱✰❱ñ✲✍✧ñ✍ò❩⑧✖✳ õ♠÷ ø✷② ➡✵❫➠❡ ⑦❩⑥✧ñ✢✴➈★✵ò❩ó✷✶♦ñ☞⑤▼⑦❩ø✞ó✸✍➻ó✷✏ õ ó✖✍ ➅➄÷✺✹ ✻✽✼ ✾✛✿ ❀❂❁★❃✘❄❆❅✸❇✝❈❊❉✖❄✱❋❂●ó❱ò✮❍❨⑥◆⑧❇⑦ ✵⑧➏ò✐⑦❩ø✏⑤✁④✺✳✷⑧➠ò ✵ò❩ó✖■✺✳✏ñ✲❏ ❧ ✰➏ø✎❑➏ñ➛⑦❜⑥◆ñ➂②♦⑦❩ò❜ó✸✍✺✰▲✏⑩ó➏ò✟❏q➛❝✷➝➼õ✠÷ ⑦❜⑥✧ñ ✴Ñ➈ ✵ò❩ó✷✶♦ñ☞⑤▼⑦❩ø✞ó✸✍➻ó✷✏ õ ó✖✍ ➅➄÷✸▼ ◆P❖❘◗ ❄❚❙ ❯❱❅❖✙❖✺❲★❖✺❲ ✴Ñ➈❨❳❖❅✝❩❭❬❪❄❳❄✷❅❆❫✑❴❛❵ ❖❘❜❳❞❝❢❡✥❣✛❤◗ ❈❖❳❫✑❴❥✐ ðó❦■✚ñ❧✰➏ø✎✍✙♠✺❍✘ñ✛✍◆ó➠⑦❜ñ✈⑦❩⑥◆⑧❇⑦✮✏⑩ó❱ò❚⑧✷✍♦♥ Õ❚÷③➃✦➅➄÷ ♠ ✉Ñ❹ õ♠÷➼öÑÕ❚÷ ❹ ❺❻❨❼✙❽✞❾✚❿ Ö ❥♠❧⑩õ✠÷✈ö❏Õ❨÷✚♣❜õ♠÷➛ö❏Õ❚÷❅q ❧ ⑤✁ó✔ñ✁ò✙⑤✆ø✎❑❅ø✞⑦✁♥q ➚ ❥♠❧⑩õ✠÷✈ö❏Õ❨÷ ➐ ❧✹õ➄ö❏õ♠÷❅q▼♣❜õ♠÷✈ö❏Õ❚÷❱q ❧ó❱ò✐⑦❩⑥✧ó✖✰❱ó✖✍◆⑧✖✳✞ø✞⑦✁♥q ➚ ❥♠❧⑩õ➄ö❏Õ❚÷✚♣✐õ✠÷➛öÑÕ❚÷✔q Ö Ò❦❹ õ➵ö❏Õ÷♠❹ ❻❼ ❽✞❾✚❿ ❹ õ÷ öÑÕ÷✵❹ ❻❼ ❽✞❾✚❿ ❧ ⑤✁ó✖✍❅⑦❜ø✎✍✔④✧ø✞⑦✁♥q ➶ ①