Note 4 Continuity of v in X It is clear that if v E Xh, then since Xh CX(X ember of Xn is a member of X because Xh Z vEX+ggg u(Mz u(ma M aaamembers of v in t h(Q2(anr hence Xh CX anish at l z Manr la nWt Xn CX also tells us that v must be co X uous Tthe (r istributionalreriatiwe of the function r epicter abo"e is piecet ise constant on each element, anr hence are integrable, as re Wuirer by t h(s thot e"er, if t e har almps in v bett een elements, the reri ati e t oulr generate relta r istributions at the nor es, t hich are Xoq in i r(Q-(see rote b of the last lecture thus Xo gin t h(s rIt is important e that t e ro not re Wire that our u be in Ch(Q, that is, ha"e continuous Tst reriaties-this is much more r iy cult to implement numerical We X - these are knot n as Xo Conforming approlimations, as opposer to tI conforming appro imations t e consirer herent Y, set of members pHE Y, y a nQ.Si is a basis for y if anr only if ∈Y,miWe= HE F such that p a=HpHi rim(ension(Y M n 回n回 It follot s from our re nition of a basis that any set o members pH- members such that HpHa Mt Hr my a nQ. SM t ill ser e basis it is al ily choice of basis is not at all uniMue, the rimension of Y, rim(y, w uniwi pr simplicity t e t ill use the basis concept primarily in the contel t of miter imensional spaces such as Xnubut in mite r imensional space t h(@ can also be rescriber in these terms mr ote tÐ❼Ñ❲Ò✑Ó❴Ô Õ✥Ñ❣Ö❝Ò✣×❆Ö▼Ø✞×❉Ò❆ÙÚÑ✛Û⑦Ü✙×✰Ö↕Ý❨Þ ß❵à✆á✁â✼ã✦ä✓å☛æ✡çèà✟é✕æ❥à✆á✧ê➌Ü❨ë★ÝÞ✛ì à✣é✕å✢í✹â❉á✓í✕ã✢å❀ÝÞ❤î Ýðï❆ÝÞ á✓â✼æ❖ñ✑ò✕ó✦ñ✰ô✕õ✡ö☞÷❄ø✒ê➫Ý➼ùúæ✒í❣û üåü❩ýå☛çþø✡ê✞Ý❨Þ✸á✓â➌æ üåü✥ý å✢ç❝ø✡ê✕Ý ý å☛ã☛æ✒ÿ✕â✣å❈Ý✬Þ✁￾✄✂☛Ü✲ë❴Ý✆☎✞✝✟✝✠✝☛✡✌☞➣Ü❙ï✎✍✏☞✑￾✿Ü❙ï✓✒✔☞✑￾✕✍✼ù õ✞✖✗✖ üåü❩ýå☛ç✟â❝ø✡ê✞Ü❄á✓í✙✘✛✚✜ ï✣✢✤☞✼ï❆æ✡í✦✥❩é✛å✢í✕ã✢å✼Ý❨Þ î Ý✧☞✩★❥æ✡í✛á✓â✣é✬æ❥à✫✪✬￾✕✍Pæ✒í✦✥✁✪✬￾✆✒✏✭✯✮❈ÿ✽à Ý❨Þ î Ý æ✒ä✁â❉ø❩à✣å☛ä✧ä✁â❈ÿ✕â❈à✣é✞æ❥à⑦Ü üÿ✕â❉à ý å❨ö✱✰✞✲✦✳✣✴✗✲✕ò✦✰✒ò❣ñ✶✵þà✣é✛å✬ï✎✥✽á✓â❉à✣ç✟áýÿ✽à✣á✓ø✡í✞æ✒ä✗☞✑✥✽å☛ç✣á✷★❥æ❥à✟á☛★✄å ø✒ê❜à✟é✛å✆ê❛ÿ✛í✞ã✑à✣á✓ø✡í✸✥✛å✠✹✛á✁ã✑à✟å✟✥❨æýø✞★✄å✼á✓â✑✹✛á✧å✎ã✦å✟✺✼á✓â✣å❬ã✦ø✄í✕â❉à✟æ✒í✴à▼ø✄í✬å☛æ✡ã☞é❤å☛ä✧åüå☛í✴à ì æ✒í✦✥❤é✛å☛í✕ã✦å â✼✻✄ÿ✞æ✒ç✟å✼á✧í✴à✣å✟✽✡ç☞æýä✓å ì æ✡â❝ç✟å✟✻✴ÿ✛á✓ç✣å✔✥ ýû✾✘✛✚✒ï✎✢✤☞❀✿✴é✛ø✞✺❈å✠★✄å✢ç ì áê❁✺❈å✆é✕æ✏✥❃❂➴ÿü✹✕â▼á✓í✬Ü ý å✦à❄✺❈å✢å☛í å✢ä✓åüå☛í✄à☞â ì à✣é✛å❅✥✽å☛ç✣á✷★❥æ❥à✣á✷★✡å❆✺❈ø✡ÿ✛ä❇✥❈✽✡å✢í✕å✢ç☞æ❥à✣å❅✥✽å☛äà☞æ✾✥✽á✓â❉à✣ç✟áýÿ✽à✣á✓ø✡í✞â✆æ❥à⑦à✣é✕åPí✛ø❉✥✽å☛â ì ✺✼é✛á✁ã☞é æ✒ç✟å❊✲❋✰●✳❃á✧í■❍❑❏✡ï✎✢✤☞❨ï❆â✣å✢å▼▲❬ø✒à✣å❈◆★ø✒ê❈à✣é✛å❨ä✁æ✡â❉à✥ä✓å☛ã✦à✣ÿ✛ç✟å✔☞✸ù æ★ê❛ÿ✛í✕ã✦à✣á✓ø✡í❖✺✼á✧à✣éP❂➴ÿü✹✕â✥á✓â à✣é❣ÿ✕â◗✲❋✰●✳❝á✧í❈✘✛✚❥ï✣✢✤☞❀✭❝ß❵àèá✁âèáü✹❙ø✡ç✣à✟æ✒í✴à▼à✟ø✥í✛ø✡à✣å❬à✣é✕æ✒à❑✺❈å❘✥✽ø✥í✕ø✒àèç✟å✟✻✴ÿ✛á✓ç✟å✼à✣é✕æ✒àèø✄ÿ✛ç➲Ü ýå á✓í▼❙❚✚❥ï✎✢✤☞ ì à✟é✕æ❥àèá✓â ì é✞æ✌★✡å✼ã✦ø✄í✄à✟á✧í❣ÿ✛ø✄ÿ✕â❱❯✕ç☞â❉à✑✥✽å✢ç✟á☛★❥æ✒à✣á✷★✡å☛â➫ùðà✣é✛á✁â▼á✁â üÿ✕ã☞é üø✡ç✟å❲✥✽á❨❳❨ã✢ÿ✛äà à✣ø❤áü✹✛ä✧åüå✢í✴à❬í❣ÿüå✢ç✟á✓ã☛æ✒ä✓ä✧û✏✭ ❩❼å❻ç✟åüæ✡ç❭❬↕à✣é✕æ✒à❖à✟é✛å✢ç✟å❻æ✡ç✣å❪❯✕í✛á✧à✣å❼å✢ä✓åüå✢í✴à✹æ●✹❫✹✕ç✣ø✌❴✽áüæ✒à✣á✓ø✡í✕â❨á✧í❵✺✼é✛á✓ã☞é⑥ÝÞ❜❛î Ý❱ù à✣é✛å✎â❉å✹æ✡ç✣åP❬❣í✛ø✞✺✼í↕æ✄âP✲❋✰●✲❙ã✢ø✡í✽ê❛ø✄çüá✓í❫✽❺æ●✹❫✹✛ç✟ø✌❴✽áüæ✒à✣á✓ø✡í✕â ì æ✡â❤ø✏✹❫✹❙ø✄â✣å✟✥ à✣ø❼à✣é✛å ã✦ø✄í✽ê❛ø✡çüá✧í✦✽❑æ✶✹❫✹✛ç✟ø✌❴❣áüæ❥à✟á✧ø✄í✕â❝✺èåPã✦ø✄í✕â✣á✷✥✽å☛ç✼é✛å✢ç✟å✶✭ ❞❢❡❤❣❋❡✷✐ ❥❧❦❫♠✔♥❤♠ ♦❋♣rq✗s✩t✈✉✶✇ ①÷②✲❜÷②③✟õ✞✖⑤④✄÷❤⑥✑✲✦✴✗✳✣✴❤✰✞✲❋✵ ✽✄á☛★✄å✢í★æ❩ä✓á✓í✛å☛æ✡ç✆â✓✹✕æ✄ã✦å⑧⑦ ì æ❑â✣å✦à✆ø✡ê üåü❩ýå☛ç✟â❵⑨✌⑩Pë✬⑦ ì✾❶ ￾❷✒✏❸✠❹✟❹✠❹②❸✱❺ì á✓â✥õ ýæ✡â✣á✓âèê❛ø✡ç✆⑦ á✧ê❝æ✒í✦✥❴ø✡í✕ä✧û❨á✧ê ❻ ⑨✲ëP⑦ ì❽❼ ÿ✛í✛á❇✻✴ÿ✛å❿❾⑩ ë❖ß➀➮â✣ÿ✕ã☞é❴à✣é✕æ✒à ⑨✙￾➂➁➃ ⑩❭➄ ✚ ❾❁⑩✫⑨✌⑩✙✿ ✥✽áüï❛å✢í✞â❉á✓ø✡í❽☞❃ï✎⑦✁☞➅￾❵❺➆✭ ▲❃➇ ▲❘➈ ➉❲✒ ➉❑➊ Ð❼Ñ❲Ò✑Ó➌➋ ➍➲×✰Ö▼Ó✶➎❉➏P➐þÓ✔➑❬Ó✒Ö➅➐þÓ✒Ö➅➒✄Ó❈➎✽Ö✑➐➓➐❜×✎➔✺Ó✒Ö⑤→✎×✉Ñ✽Ö ß❵à➫ê❛ø✡ä✓ä✧ø✞✺✆â❫ê❛ç✣øü ø✄ÿ✛ç✩✥✽å✠❯✕í✛á✧à✣á✓ø✡í❄ø✡ê✕æ ýæ✡â✣á✁â❜à✣é✕æ✒àþæ✡í✴û❃â✣å✦à➣ø✡ê❽❺♥ä✓á✓í✛å☛æ✡ç✣ä✓û❬á✓í✦✥✽å✟✹✞å☛í✦✥✽å✢í✴à üåü❩ýå☛ç✟â➣⑨⑩ ù üåü❩ýå☛ç✟â✆â✣ÿ✕ã☞é❖à✣é✕æ✒à ➃➁ ⑩❭➄ ✚ ❾⑩ ⑨⑩ ￾↔✍➙↕➛❾⑩ ￾↔✍❫❸ ❶ ￾✆✒✶❸✟❹✠❹✟❹②❸✼❺ ù➜✺✼á✓ä✓ä❀â✣å✢ç✼★✡å✹æ✡â❖æ ýæ✡â✣á✁â✠✭➥ß❵à❖á✁â✲æ✡ä✓â✣ø❺ç✟å☛æ✶✥✛á✧ä✓û✈✥✽åüø✡í✕â❉à✣ç☞æ❥à✟å✟✥↕à✣é✕æ✒à ì æ✡äà✟é✛ø✡ÿ✦✽✡é✺ø✡ÿ✛ç ã☞é✛ø✄á✓ã✢å❴ø✡ê ýæ✡â✣á✓â❤á✁â✬í✛ø✡à✲æ❥à❨æ✒ä✓ä✆ÿ✛í✛á❇✻✴ÿ✛å ì à✟é✛å➌✥✽áüå☛í✕â✣á✧ø✄í ø✡ê❃⑦ ì ✥✛áüï❤⑦❧☞ ì ✴✁ñ✹ÿ✛í✛á❇✻✴ÿ✛å✶✭ ➝✛ø✄çPâ❉áü✹✛ä✧á✁ã✦á✧à➴û❪✺❈å❧✺✼á✧ä✓ä▼ÿ✕â✣å❑à✣é✛å ýæ✄â❉á✁âPã✦ø✄í✕ã✦å✟✹✽à❃✹✕ç✣áüæ✒ç✟á✧ä✓û✹á✧í❼à✟é✛å✬ã✦ø✡í✴à✟å②❴❣àPø✒ê✑❯✞í✛áà✟å②➞ ✥✽áüå✢í✕â✣á✧ø✄í✕æ✒ä⑦â❭✹✕æ✡ã✢å☛â✥â✣ÿ✕ã☞é æ✡â❄ÝÞ ✿ ýÿ✽à❑á✓í❉❯✕í✕áà✟å✬✥✽áüå✢í✕â✣á✓ø✡í✕æ✡ä❈â❭✹✕æ✡ã✢å☛â❩â❉ÿ✕ã☞é↕æ✡â❩Ý➟￾ ✘✛✚✜ ï✎✢✤☞✼ã☛æ✒í★æ✒ä✁â❉ø ý å➙✥✽å☛â✟ã✦ç✟áýå✔✥❖á✓í❴à✣é✛å✎â❉å❃à✟å✢çüâ✠✭✑▲❬ø✒à✟å❅✺èå❄ã☛æ✒í❴å②❴❉✹✛ç✟å☛â✟â✼æ✬â✓✹✕æ✄ã✦å❅⑦ á✓í ➈