without any numerical calculation). Of course this will not be generally true for more difficult problems. e(un) put is th °(u)-C°(u Recall that e=u-Whi note the second step above follo ws from linearity of eo(u) 1.7.2 General result SLIDE 31 IfeO∈H-1(9) le(e)l scllelH()(boundedness) CO∈L2(),ther (e)l sc llellza(s)(boundedness) In fact: for any eoEH-(Q) CO(e)|≤Cl|ll a(o, vl) yv∈X a(u, vn) eo(u) Note 8 Role of adjoint (Optional) The variable a above is denoted an adjoint, or dual, variable, though we will not dwell on the meaning of the term here. We note only that the strong form for y is -vaz="eo, which can be vie wed as our original equation but now with the output functional as data; in the case of a nonsymmetric operator, y by the dual, or transpose, operator, in which the signs of e odd derivatives would be flipped relative to the original equationá✐â✁ã❁ä◆å❲æ✾ã✽ç✬è➔é➾è✰æ✧ê❬ë✫ì❁â➧í✙ç❫î◆í✙ç❫î♥í✙æ◆î♥ç✬ã❁â♥å❫è✄ï❆ð➨ñ✆ò➐í✙å❫æ◆ì✩ó❢ë✳ã❨ä◆â♥ó⑥á✐â✁î♥î✧è✧å✬ã❇ô❅ë✼õ❫ë✫è◆ë✙ì✩ç✬î♥î✁é➳ã❁ì❨æ◆ë✐ò➥å❫ì ê✤å❫ì❨ë❀ö✾â❈÷✹í✙æ◆î❈ã✼ø✧ì❁å❲ô◆î✁ë✫êPó✙ðùï ú❅û✰ü✮ý❏þ▲ÿ✁￾ ✂☎✄✝✆✟✞✡✠☞☛✍✌✎☛✑✏✑✠✑✒✔✓✖✕✘✗✚✙✜✛✣✢ ï✥✤✧✦✙✩★ ✓✫✪✬✕✔✗✙ ✛✣✢☎✪ ï✭✤✧✦✙ ✮ ✯✍✰ ✱✳✲ ✴✫✵✫✶ ✷✹✸✺✸✼✻✽✸✼✾✩✸✼✵✚✷✥✿✫❀❁✸✼❂✫✶✽❃☞✷✹✶✽❄✑✵✖❄❆❅✳❄✑❇❈✷✹✿❈❇❈✷ ë✙ì❨ì❨å❫ì❝â✁è➷å❫æ◆ã❁ø◆æ✾ãsâ♥ó✳ã❨ä✰æ✄ó ❉ ✓✜❊❋✓ ✪ ❉ ✕ ❉ ✗✙ ✛●✢ï ❊❍✗✙ ✛●✢✪ ï ❉ ✕ ❉ ✗✙ ✛✣✢■❊❏✢ ✪ ï ❉ ✕ ❉ ✗✙ ✛●❑ ï ❉☎▲ ▼☛❆◆❆❖◗P✟P✁✠✣❘✳❖❙✠❚❑✖✕❯✢❱❊❲✢❳✪✁❨✥✞❬❩❙✠❭☛❪✠✟❘✡☛❫✏✚☛❆◆✎❩❙✞❬❴❵✏✑✠☞☛❭❛❜❖❞❝❆❩◗❡◗☛✥✄✍❩❙P✟P❢❩◗❣❤✏✐✄❆✌❥❩❙❦❧P♠✆✟✞❬☛❆❖◗✌✑✆✟✠✹♥♦❩✼✄♣✗✚✙❪✛●q ïsr t❬✉✇✈✐✉✇① ②④③⑥⑤♣③⑧⑦✫⑨✡⑩❷❶❸③⑧❹❈❺♣⑩✣❻ ú❅û✰ü✮ý❏þ▲ÿ☎❼ ❽ò ✗✚✙❿❾❜➀❋➁✐➂✫✛❁➃ï ★ ã❨ä◆ë✙è ❉ ✗✚✙❪✛❁❑ ï ❉⑥➄✔➅➇➆ ❑ ➆✍➈✩➉✎➊❢➋❚➌ ✛ô✄å❲æ◆è✧ö✾ë❛ö✾è◆ë❛ó❁ó✩ï✖ð ❽ò ✗✚✙❿❾❜➍❷➎❙✛❁➃ï ★ ã❁ä✧ë✙è ❉ ✗✚✙❪✛●❑ ï ❉✁➄✘➅➇➆ ❑ ➆✚➏✳➐❆➊❢➋❚➌ ✛ô✄å❲æ◆è✧ö✾ë❛ö✾è◆ë❛ó❁ó✩ï✖ð ú❅û✰ü✮ý❏þ▲ÿ✳➑ ❽è➷ò✘ç❫í❆ã ✒ ò➥å❫ì✼ç✬è➔é ✗✚✙✘❾➒➀❋➁☎➂❙✛❁➃ï ★ ❉ ✗✚✙❪✛●❑ ï ❉✁➄➓➅➔➆ ❑ ➆✍➈➉ ➊→➋✡➌✺➆✑➣ ❊ ➣✪ ➆✚➈➉ ➊❢➋✡➌ á✐ä◆ë✫ì❁ë ↔ ✛✣q❬↕ ➣ ï ✕ ❊❫✗✙ ✛✣q ï ↕ ➙✬q■❾❸➛ ↔ ✛✣q❬↕ ➣✪ ï ✕ ❊❫✗✚✙✩✛✣q ï ↕ ➙✬q■❾❸➛✪ ↕ ➜✖➝ ç✬è✄ö ➣ â➧ó✼ç✬è✡ç❫ö◗➞❃å❫â♥è❲ã ★ å❫ì✼ö✾æ✄ç✬î ★✳➟ç✬ì❨â♥ç❫ô◆î✁ë❲ð ➠❏➡☎➢✑➤❜➥ ➦❜➡✁➧☞➤➒➡✳➨❪➩❬➫✫➭⑧➡⑥➯❁➲✺➢❸➳❆➵❵➸❪➢➺➯✼➡⑥➲❤➩✁➧✣➻ ➼ä◆ë ➟ ç❫ì❁â➧ç✬ô✧î✁ë ➣ ç❫ô✄å➟ ë❝â♥ó✽ö✾ë✙è◆å❫ã❁ë❛ö❬ç❫è✹ç❫ö✫➞❃å❲â✁è✰ã ★ å❫ì✽ö✾æ✧ç✬î ★⑧➟ç❫ì❁â➧ç✬ô✧î✁ë ★ ã❁ä◆å❲æ◆õ❫äPá✳ë✐á✐â♥î✁î è◆å❫ã✆ö✾á✽ë✫î✁î❏å❫è➷ã❁ä◆ë❀ê✤ë❛ç✬è◆â♥è◆õPå✬ò➞ã❁ä✧ë➫ã❁ë✙ì❨ê➡ä✧ë✙ì❨ë❫ð➾➽▲ë❀è◆å✬ã❨ë❀å❫è◆î♥é✹ã❨ä✧ç✵ã✐ã❨ä◆ë✶ó❢ã❁ì❨å❫è◆õ❬ò➥å❲ì❁ê ò➥å❫ì ➣ â➧ó ❊ ➣✺➚❈➚ ✕➶➪✚❊❫✗✚✙❪★ ➹ á✐ä◆â➧í✩äÜí✙ç✬è✡ô❅ë ➟ â✁ë✫á✽ë❛ö➣ç❲ó✐å❫æ◆ì✐å❲ì❁â♥õ❫â♥è✧ç❫î❅ë❈➘✰æ✧ç✵ã❨â✁å❲è➣ô◆æ◆ã✆è◆å✵á á✐â✁ã❁äÜã❨ä◆ë➳å❫æ◆ã❁ø◆æ✾ã✆ò➥æ◆è✧í✖ã❨â✁å❲è✧ç✬î❡ç❫ó✆ö◆ç✵ã✩ç ✲ â♥è✡ã❁ä✧ë➾í✙ç❲ó❢ë❀å❫ò❇ç ✞❬❩❙✞✳✏✑♥◗❦➴❦➷☛✚✠❁✌✑✆●◆ å❫ø❅ë✙ì✩ç✵ã❁å❲ì ★ ➣ á✳å❫æ◆î➧ö➷ô✄ë➳ç❲í✖ã❁ë❛ö➷æ◆ø✄å❲èÜô➔é✹ã❁ä✧ë➳ö✾æ✧ç✬î ★ å❲ì❝ã❁ì✩ç✬è✧ó❁ø❅å❲ó❁ë ★ å❫ø❅ë✙ì✩ç✵ã❨å❫ì ★ â✁è✡á✐ä◆â➧í✩ä➷ã❁ä◆ë➾ó❢â♥õ❫è✧ó❝å❫ò ã❁ä✧ë❀å➔ö✧ö➷ö◆ë✙ì❨â➟ ç✵ã❨â➟ ë✫ó✳á✳å❫æ◆î➧ö✟ô❅ë✝➬✧â♥ø◆ø❅ë✫ö✡ì❨ë✙î➧ç✵ã❁â➟ ësã❁å✤ã❨ä◆ë❀å❫ì❨â♥õ❫â♥è✧ç✬î✿ës➘✰æ✧ç✬ã❁â♥å❫è➐ð ➮✫➱