Linear space. A set Y is a linear (or vector ) space vu1,v2∈Y, va∈R,u∈Y, aU∈Y Linear forms, L(u) L:、Y.,→、R,( form or functional) L(av +U2)=al(on)+l(u2)(linear) Va∈R,V Bilinear forms, B(w, v) B(w, t)linear form in w for fixed D B(U, u)linear form in v for fixed w(bilinear) Note that B:YxZ,R indicates that b has two inputs (arguments the first from the space y, the second from the space Z; the output is a real number. SLIDE 15 B(U,U)=B(U, w) SPI B(,w)>0 ≠0SPD 2.3.3 Restatement SLIDE 16 a(w, v=ÝßÞ❨à✭á✴â❀ã♣ä●å✥â✢æ✴á✏çPè❴é ê➫ë✎ì✶í èïî ë✤ð❡ñ î✕òì❚ð✦ó✈ô✛õ✦ó✤ö✢ì❚÷✶í✎õ✦ó❍ø♠ë❊ùPð✦÷✶ì î♥ú û✒ü✢ý❀þ✎ü✺ÿ✁￾ èþ ü✢ý✄✂❝ü✺ÿ☎￾ è û✝✆✞￾✠✟✡✒þ û❦ü☛￾ èþ ✆▲ü☞￾ è ✌ ✍✏✎✒✑✔✓✖✕✘✗✚✙ ÝßÞ❨à◗á✴â✺ã✜✛✣✢❀ã✥✤✙ä✢ç✧✦ ôü ø é ✦❈é è ★✪✩✣✫✪✬ ✭ ✮✰✯✲✱✣✳ ✴ ✟✡ ★✪✩✣✫✪✬ ✵ ✱✣✳✶✯✰✱✣✳ ô✛✣✢❀ã✥✤ õ✦ó ✛✸✷✠à✭æ✣✹●Þ✶✢❀à✭â✻✺ ø ✦ ô✆▲ü✢ý✜✂❝ü✺ÿ ø✜✼ ✆✦ ôü✢ý ø ✂ ✦ ôü✺ÿ ø ô ✺❱Þ❨à◗á✴â❀ã ø û✝✆✞￾✠✟✡❲þ û✒ü✢ý✺þ❊ü✺ÿ✁￾ è✾✽ ✍✏✎✒✑✔✓✖✕✘✗✻✿ ❀❞Þ✔✺❭Þ❨à◗á✴â❀ã❁✛✣✢❀ã✥✤✙ä✶ç✧❂ ô✶❃ þ❊ü ø é ❂☎é❞è❅❄❇❆ ✴ ✟✡ ô✛✣✢❀ã✥✤ø❉❈ ❂ ô❊❃ þ ü ø♠ñ î✕òì❚ð✦ó úõ✦ó●❋ î♥ò ❃ úõ✢ó■❍❑❏✠ì✰▲ ü ç ❂ ô❃ þ✎ü ø♠ñ î✕òì❚ð✦ó úõ✦ó●❋ î♥ò ü úõ✢ó■❍❑❏✠ì✰▲ ❃ ô●▼ Þ✔✺❱Þ❨à◗á✴â❀ã ø ✌ ◆❖✢P✹❪á❉✹❊◗✥âP✹❘❂❺é✤è❙❄❚❆ ✴ ✟✡ Þ❨à✏❯✺Þ✽æ✴â✻✹✐á❍ä❉✹✔◗✸â✻✹❘❂❱◗✥â❀ä❲✹❨❳✄✢❦Þ❨à✺å✏✷❩✹✻ä❖❬★â❀ã❪❭✚✷❩✤❴á✶à❫✹✽ä❵❴❜❛❝✹❊◗✥á❡❞❖ã✴ä✸✹ ✛✴ã❢✢P✤❣✹❊◗✥á✒ä●å✥â✢æ▼á❤è❤❛✜✹❊◗✥á✒ä✏á✴æ●✢✺à✏❯✐✛✴ã❢✢✻✤❥✹❊◗✥á✒ä✻å✸â✦æ✴á✁❆✁❦✜✹❊◗✥á❧✢✻✷❩✹❱å✏✷❩✹❖Þ✰ä✙â❴ã✎á✴â✻✺❒à✧✷❩✤▼á✏ã✲♠ ✍✏✎✒✑✔✓✖✕✘✗✚♥ ♦❩♣✜q❱rî ñ î♥òì⑨ð✺ó úõ✢ó❢❋❴ë ç❑❂ ô❊❃ þ❊ü ø é ❂☎é❞è❅❄❋è ✴ ✟✡ î ës▼ Þ✔✺❭Þ❨à✭á▼â✺ã ❈ ❂ ô❊❃ þ✎ü øt✼ ❂ ô üPþ ❃✝ø ♦✈✉❁✇ ❈ ❂ ô❊❃ þ ❃✝ø②①④③ ç û ❃ ￾ è➞ç ❃❅⑤✼⑥③✾⑦♣⑧q ✌ ⑨❘⑩❷❶✖⑩❷❶ ❸❺❹❼❻✲❽✪❾❩❽✰❹❼❿➀❹❼➁➂❽ ✍✏✎✒✑✔✓✖✕✘✗✚➃ ➄ì✏í ➅ ô❊❃ þ✎ü ø❁✼➇➆❼➈⑥➉✝❃⑥➊✰➉ü❲➋✒➌❝þ û ❃ þ✎ü☞￾➎➍ ➏