Question

3) Let (en) and (tn) be two orthorormal bases for Learb). Let it be the space of square-integrable functions of two variables on the square a sx,y<b, with inner product $$ f(x,y) 96,4) dedy. a) show that the set of functions e; (x) J. (y) is ortho normal in H b) show that if d et and Soxxy) e; (x) f; (4) dxdy =0 Hij then f=o. c) The set of functions eix) d; (y) is labelled by two integers and is therefore countable, and can be arranged in a sequence. prove that this sequence is a basis of Ho

Answer 1

Solukom i tems and fm3 befwo withondarmee baut fort(06) s-laib) H = He space of square Irlogstable fundions of two variable Oon the square axis Innost product tegemiyd> = "1" filmu) geng) dxdy, Ja to where togony) e H 0 H = {ti Chilli falary) - filmiyla is arhonmal from H if utene 2. Inne proguer lily 11 = 1 , .. 11 11 moim Iach vectar of His unit kostar and His withogrtal et of vector fi 604 446,41% =0 where it 04 för.43.96. 47 = fals4gzodudy. = 0 where itý 10 of function ():(y) is whermomal H If QGH, Cila) fyly) is set of function defined inner product chook on phonormal Qay), Cinofily > > b bwy) eilad fyty) dady ir aray = $us 2. gram @ray andy - plyn by emul’andy = 1. umit marm again denu) ea exfiy 11 = 1 is also umit moymiis toch wee fall of H i dory), eitud 1999 = 141413) ezen) Ajlyandy = 0 wrong {qeniyl cim fý (y) 4 = 0 where ist } upy ei and ty so hormonal and eiln) fil4l is orthormeel in H. Bit den and be bormy eithesly dudy=0 Viit we know that inner product Lab), Eit fit4>= 1o ) eine;f4exdy whou eie:19 pm no Real Valued furthon = 16 lamy) ) ejiyidx dy is ciim ficy) is arthamonmal in H Thon Do 16 Qomy) eiune:14)=0 where itº By authonemal propertief I bpbdemy) eitweity) = 0 bwig if only 64). Ödudy = open { diniy o>'- then of = 0 084 {oi eilne fulg?) = lo enl fulg) andy Them f=oph © The set of function ein fjlyl is two intrynable function Eveng orthonormal sel of vectore is un curly indent Then fegaonce comment is a basis of the