ft, product ,y>=o the As a Still L by > Over ( x,y > = consequence, not vectors are orthogonal A ( x too : say products provide aisles, Inner Inner

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1 As a consequence the norm determines the Inner product Inner products provide aisles too : over R : 4 L by > ft 1 64 # o ) SQ x y > coso! ) Over Still Q not so dear sauce % y > is conplex : Us gy ) 0 is special cos G 0 o We say vectors are orthogonal A x y>o

2 sen Vectors ei en MX are onthonamt if He lt#vi eies > O itj La ej > Sig) [ Eu je Lanna : Any finite collection of orthonomat vectors is land Pf : Suppose 4Gt n then 0 Then q+ 04 ejek > teen ek > cj< CK for each k Here e " e one a basis for their span If es span a ) Gelt " Chen * e < 7 ck

3 That is it I give you x know you ck via < x q > Compare : 1W 421 ) wz was 23 5) 3 ) * a an tt :] Starkwla fifty 14 Thun : Every finite dimensional vector basis : admits an o n basis Pf : Gram 7 Xi e Xn I Schmidt space fz Xz xz ez f2 Hfd ) e fz e ) e xz > q Lee > e

4 < vectors ' Lin x fktxk xk e > e k ) Ei > 9<1 ekfk/hfhk At each stage fkt 0 else xk is a liner combo of e % ad of ) and ek e ; S 0 k > j Result is n on isaba iwdjvightcaant

5 Consular the corfiguutrdi The vectors in Y are all onth to X Y ' \ and indeed it is all of such vectors Def : Given AEX any subset the orthogonal omrbrut ottis : At { x X : x a > to tae A AEB } A A$ we the At X ) Observations 1) At the ortho complaint of A) is a subspue even A A isiot 2) 0 At always 8) If Bte At gnats a subspace 4) At )t2a but need at guy A

6 5 At is closed I xnin At n x Fix aea dam < x a > kn< xn a > 0 In fact ^ x 4n 4 < in y > < x y> L a yn > is ) 1<44^54 y > + 4uy > Gif e Kxnyry> + lcxn x y > ) E Hall Hynyll + Ha ) If A contains an open ball At 0 04 a + If yett yt 0 alt Br a) a A ) EE * EA n ya > the "dp E 14/1? oops!

7 Distance from a set dp A) This distance always exists naff dlp a) But it may not be thut7aaydlpallpdesfjal@pdlpa ) 0 but if dlp a) to a p Sometimes closest points exist but are not unique YD Fault of the weadshpe

8 But also : D) % am an these Points are distance 1 fan p Result : if AEX and iscauex closed and X is a Hilbatspacg the closest points exist and we unique Def A EX is come if for any pqet l E) p + tq e A ttesgi ] : #tl p A to

9 The three examples show not closed or rot area or not Hilbert and the resort can fail Thu : If A is a closed convex subset of a Hilbert exists a unique space X gun xex three aea dla 7dAxb pf : Let an be a sequence in A d an x) d a x ) dl " At d Math By the panllelosm law H a Now m 11 p an ) )lp +111Pa p Ip an ) + { 211 am )H2 pam/l2t2hpanh2 + lkpa) kpam) pla#en p ante ) P 3 4d p A) sweet is Coward Thus Han am/pe4ldlpa))+tn+tm4dlpa)

10 4Hp I a) Hence { an } is Candy and conages to a lmita Since A is closed a A Moreover d p A ) s dlp a) 1in d p an ) dlp A) This establishes existence For Uniqueness if a and az ae minimizes Hd + a) p a) lp a) * p 1/2 { a + 24paz//2 p Ie pae a±e)h2 + Ha adpe4dlp A) sohuhr 10

11 Next ap : z 7 y*z * + y we x Given a subspace Y and E decompose yt y + z ye4 would like to Z e Yt This doesn't always work : ZE l " Z 't { o } If and w Zt z Z # w 0 so ztw ZGZ The key extm msnat is that Y must be closed

12 < &<y YH Before we start : Yt z* * W \ vs Hz ylp {? conpme s W! ) Lanna : If ze Yt then Hzlk HZ fan ally EY Pt : Hz y z y > z > 11* ' L 11211? onuesi If < for all yet z e Yt Hz a Lzy > so Re % if > + Raptly 112 EHHF

13 Pick y so z y > FO and pick x so z y ) do ) Then for D Ea t > o t2 1% s Zt But then <24 k 20 nd a contradiction

14 all Since a Thn : If Ye X is a closed subspace of a Hilbert given ex there exists YEY and 2 e Yt spug a unique YTZ Pf : Let XGX Y is closed al auex there exists ye Y dlqy )d x Y ) Let z x Y I claim zey Indeed if at Y Hz H * y dhµy 1/3 ) Thus ze Yt As for uniqueness : ZX Y + Z Yzt Zz 1241 Z Zz EY Eyt

15 Bat ynyto < x is o! ) So 2122 Y Remark : it alb Ha+bHEHall ' t 2 a b > tsb a)o So if x ytz YEY Y's /2*112112