(u, ma-J vu Vu+uedA AllYl(s) Vu2+w2 dA N6 E4 Important theoretical and numerical implications Important spaces, inner products, and norms Hilbert and Banach Spaces A Hilbert space is a linear space Y with which we associate an inner product Dr this is simply an SPD bilinear form which then induces a norm, lolly =(u, /2. In fact, what we have just described is an inner product space: a hilbert space is a complete inner product space; by completeness we mean that any Cauchy sequence(n∈ Y such that lyn-ymly→0asm,m→ oo) converges to a member of y A Hilbert space is a special case of a Banach space Z, which is a(complete) ormed linear space. The norm z associated with a Banach space is not, in her l, induced from any bilinear form, but must still satisfy certain conditions (the conditions we intuitively associate with any measure of "length") ∈2,u≠0 lawllz al llwlla va∈R,Vu∈ l+叫ll≤‖llx+lxyu∈z,vu∈z the last being the triangle inequality(the shortest distance between two points It can readily be shown that a norm induced by an inner product automat ically satisfies the above conditions. The triangle inequality is proven with the help of the Cauchy-Schwarz inequality, which states that for an inner product (u,y)y≤ Jully lully We give the proof here 0< (u,)y (,) l =(x-2)y l lollyt✈✉✴✇✳①❇②④③⑥⑤⑧⑦✖⑨✺⑩ ❶ ❷✄❸ ❹ ❺❻❼❻✓❽❿❾✲➀✓❾✘➁✄➂❼➃✓➄➆➅ ➇ ➈⑨➊➉✉➌➋ ➉ ①▲➍➎✉✿①➐➏P➑✑➒ ➓⑧✉➔➓⑧③⑤ ⑦✖⑨✺⑩ ❶ ❷✄❸ ❹ ❻❼➁→❾✈➣ ➇ ↔➈⑨➙↕ ➉ ✉ ↕ ➛ ➍➜✉➛ ➏❱➑✾➝▲➞→➟ ➛➡➠ ➢✾➤ ➥➧➦ ➨✄➩✾➫✺➭✤➯✄➲❿➳✤➵✺➲✦➸→➺✺➻✓➼✵➽➾➻⑧➸➾➚●➪❼➶✌➹➘➳✌➵➷➴➊➬✞➮❯➱✻➻✓➽➾➚●➪❼➶✌➹❐✃❛➩✿➫❤❒❩✃✈❮❪➳✤➲♥✃✈➭✤➵✺❰⑧Ï ÐÒÑ✠Ó✄Ô♦Õ Ö✓×ÒØ✾Ñ❇ÙPÓ➾Ú❇Û★ÓÝÜ✄Ø✛Ú➷Þ✵Ô✌Ü❱ß❂à❡Û✦Û✛Ô✌ÙáØ➧Ù✵Ñ❍â★ã❊ÞPÓ→Ü✵ß★Ú❇Û✛â➡Û➧Ñ❇Ùä×➎Ü å❣✃❛❒●æ➾ç✓➯✄➲⑥➳✤➵➷➴áè✾➳✌➵➷➳❱❮➾é✢ê✌➫✺➳✵❮❪ç⑧❰ ëíì➚✖➹●î➷➻✓➽→➸ðï→ñ✺➶✵➪✓➻⑥➚●ï➘➶✻❒❩✃❛➵➷ç❪➳✤➯✾❰♥➫❯➳❱❮➾çÝòôó⑥➚✖➸→➺õó⑥➺✺➚●➪❪➺õóð➻✾➶✵ï➾ï→➼✞➪✓➚●➶✌➸→➻⑥➶✌➬✢✃❛➵✲➵➷ç✓➯ö➫➷➯➾➭❼➴✌÷✺❮⑧➲ t④➋●✇✓➋ø②✳ùûú ➸➾➺❯➚●ï➊➚●ï➔ï✳➚●➱✻ñ❯➹●üÒ➶✵➬➜ý❯þ✛ÿ î✺➚✖➹●➚✖➬✺➻❼➶✌➽ ✁➼✵➽➾➱ ú ó⑥➺❯➚●➪❪➺ ➸➾➺❯➻✓➬➜➚●➬✄✂❇➮✲➪⑧➻❼ï➔➶ ➵➷➭✌➯✄➩✆☎ ➓✄✉á➓ ù✞✝ t✘✉✴✇→✉➐② ➞→➟ ➛✠✟☛✡➬ ✁➶❱➪✄➸☞☎★ó⑥➺✲➶✤➸➔óð➻➙➺✺➶✍✌❱➻✏✎④➮✺ï✳➸✑✂❯➻❼ï➾➪⑧➽➾➚✖î➷➻☞✂➜➚ ïá➶✌➬➎➚●➬❯➬❯➻❼➽➊ñ❯➽➾➼✒✂❇➮✺➪✄➸ ï→ñ✺➶✵➪✓➻✔✓▲➶ ì➚✖➹●î✲➻❼➽✳➸✴ï→ñ✺➶✵➪✓➻➔➚ ï❣➶ ❮➾➭✌➩✿➫❤❒✖ç✓➲➆ç➙➚✖➬❯➬✺➻✓➽❣ñ❯➽➾➼✒✂❇➮✺➪✄➸✴ï→ñ✺➶✵➪✓➻ ➒ î✞ü ➪✓➼✵➱✻ñ❯➹●➻⑧➸→➻❼➬❯➻❼ï➾ï▲óð➻ ➱✻➻❼➶✵➬➙➸→➺✲➶✤➸✿➶✌➬✞ü✖✕ð➶✌➮✲➪❪➺Pü➙ï→➻☞✗P➮❯➻❼➬✺➪⑧➻ t✙✘✠✚✜✛ ò✑ï✳➮✺➪❪➺ ➸→➺✺➶✌➸ ➓✢✘✔✚✏✣✤✘✔✥➔➓ ù✧✦✩★ ➶✵ï✫✪ ✇✭✬ ✦ ✮② ➪⑧➼✵➬✯✌❱➻✓➽✱✰✵➻❼ï✛➸→➼õ➶✻➱✻➻✓➱➔î✲➻❼➽⑥➼✁ ò ✟ ëûì➚●➹●î✲➻❼➽✳➸⑥ï✳ñ✲➶✵➪⑧➻Ý➚ ïð➶áï→ñ✲➻✔➪⑧➚➶✌➹❤➪✓➶✵ï→➻Ý➼✁ ➶✳✲❴➶✌➬✺➶❱➪❪➺➙ï✳ñ✺➶❱➪⑧➻✵✴✏☎✞ó⑥➺❯➚ ➪❪➺ ➚ ï❴➶ t ➪⑧➼❱➱áñ✺➹✖➻✓➸→➻ ② ➵➷➭✌➯✄➩✻ç❪➴➙➹✖➚●➬❯➻❼➶✵➽➘ï→ñ✺➶✵➪✓➻ ✟✷✶➺❯➻Ý➬❯➼✵➽➾➱ ➓ö➋✔➓☞✸ ➶✵ï➾ï✳➼❇➪⑧➚➶✤➸➾➻☞✂✻ó⑥➚✖➸→➺❘➶✹✲❴➶✌➬✺➶❱➪❪➺ ï→ñ✺➶✵➪✓➻✾➚ ï✛➬❯➼✵➸☞☎✞➚●➬ ✰✵➻❼➬❯➻✓➽❪➶✌➹✺☎✤➚●➬✄✂❇➮✺➪✓➻☞✂ ✁➽→➼❱➱ ➶✵➬Pü➊î❯➚✖➹●➚●➬❯➻❼➶✵➽ ✁➼✵➽➾➱✻☎✵î❯➮❇➸✛➱➔➮✺ï④➸➧ï✳➸→➚●➹✖➹✲ï→➶✌➸→➚ ï✁üá➪⑧➻✓➽→➸➾➶✵➚✖➬✻➪⑧➼❱➬✄✂❇➚✖➸→➚●➼✵➬✺ï t ➸➾➺❯➻✴➪⑧➼❱➬✄✂❇➚✖➸→➚●➼✵➬✺ï❴óð➻❣➚●➬❱➸➾➮❯➚✖➸→➚✼✌✵➻✓➹●ü ➶❱ï→ï→➼❇➪⑧➚➶✤➸→➻Ýó⑥➚✖➸→➺ ➶✌➬✞üõ➱✻➻❼➶❱ï✳➮✺➽→➻▲➼✁✾✽ ➹●➻✓➬✿✰✵➸→➺✄❀ ② ✓ ➓✄✉➔➓ ✸ ❁ ★ ❂ ✉❃✛ ✴✇ð✉✞❄➇ ★ ✇ ➓❆❅✦✉➔➓ ✸ ➇ ↕ ❅ ↕ ➓✄✉á➓ ✸ ✇ ❂ ❅✧✛ ✡❇ ✇ ❂ ✉❈✛ ✴ ✇ ➓✄✉ ➍➎①❤➓☞✸ ❉ ➓⑧✉➔➓❊✸❘➍❦➓⑧①❤➓☞✸ ❂ ✉❃✛ ✴✇ ❂ ①✜✛ ✴ ✇ ➸→➺✺➻❣➹●➶❱ï④➸❴î✲➻❼➚✖➬✄✰✻➸→➺❯➻Ý➸→➽➾➚➶✌➬✿✰❱➹✖➻Ý➚✖➬✺➻☞✗P➮✺➶✌➹●➚✖➸④ü t ➸➾➺❯➻✴ï✳➺✺➼✵➽→➸→➻❼ï✳➸❋✂❇➚●ï✳➸➾➶✵➬✺➪⑧➻▲î➷➻⑧➸④óð➻✓➻❼➬ ➸④óð➼➔ñ➷➼✵➚●➬P➸➾ï ✟❊✟☞✟ ② ✟ ✡ ➸➐➪❼➶✌➬❘➽→➻✔➶✔✂❇➚●➹●ü➙î✲➻✴ï→➺❯➼✤ó⑥➬❘➸→➺✲➶✤➸✾➶á➬❯➼✵➽➾➱ ➚●➬✄✂❇➮✲➪⑧➻☞✂❘î✞ü ➶✵➬ ➚●➬❯➬❯➻❼➽✿ñ❯➽➾➼✒✂❇➮✺➪✄➸✾➶✵➮❇➸→➼❱➱õ➶✤➸✭● ➚ ➪✓➶✌➹●➹●ü ï➾➶✤➸→➚ ï✭❍✺➻❼ï❴➸➾➺❯➻➊➶✵î✲➼■✌❱➻▲➪✓➼✵➬✄✂❯➚✸➸➾➚✖➼❱➬✺ï ✟❏✶➺❯➻▲➸➾➽→➚➶✌➬✿✰❱➹✖➻❣➚●➬❯➻❑✗❱➮✲➶✌➹●➚✸➸④ü ➚●ï✾ñ❯➽→➼■✌❱➻✓➬❘ó⑥➚✸➸➾➺ ➸→➺❯➻ ➺❯➻❼➹✖ñ ➼✁ ➸➾➺❯➻▲✕ð➶✵➮✺➪❪➺✞ü▼●❿ý❇➪❪➺✞ó❴➶✌➽✱◆▲➚✖➬❯➻❑✗P➮✺➶✌➹●➚✸➸④ü✠☎❤ó⑥➺❯➚●➪❪➺Òï④➸❪➶✤➸➾➻❼ï✾➸→➺✺➶✌➸ ✁➼❱➽Ý➶✵➬♦➚●➬❯➬❯➻✓➽Ýñ❯➽➾➼✒✂❇➮✺➪✄➸ t④➋●✇✓➋ø②✳ù ☎ t✈✉✴✇✳①❇②✳ù❖❉û➓⑧✉➔➓✓ù➜➓⑧①❤➓❼ù ➠ PÒ➻✏✰✵➚✼✌✵➻Ý➸→➺❯➻❣ñ✺➽→➼✞➼✁ ➺❯➻✓➽➾➻✔✓ ★ ❉❘◗◗ ◗ ◗ ✉❙✣ t✘✉✴✇→①✞② ù ➓✄①✠➓ ➛ù ① ❚◗◗ ◗ ◗ ➛ ù ➇ ↔✉❙✣ t✘✉✴✇→①✞② ù ➓✄①✠➓ ➛ù ①➷✇✳✉❖✣ t✘✉✴✇→①✞② ù ➓✄①✠➓ ➛ù ①➝ ù ➇ ➓✄✉➔➓ ➛ù ✣❱❯ t✘✉➊✇✳①❇② ➛ù ➓✄①✠➓ ➛ù ➍ t✘✉✴✇→①✞② ➛ù ➓✄①✠➓ ➛ù ➇ ➓✄✉➔➓ ➛ù ✣ t✈✉✴✇✳①❇② ➛ù ➓✄①✠➓ ➛ù ➒ ❲