Question

[8 marks] For a function space, the scalar (or inner) product of two functions f(r) and 8() is defined as (.8) = f()8(r)dr (a) Show that this definition of the scalar product satisfies all axioms of an inner prod- uct. Brief answers are sufficient. (b) Consider the functions Lo(r) =1 and L(r) =r and L2(r) =-. You may assume that Lo, L1 and L2 are an orthogonal function set, with respect to the scalar product defined above. Consider an arbitrary linear combination w(r) = alo(r)+ BL1 (r) with a, Be Rand a,B 0. Show that w is not perpendicular to Lo and that wis perpendicular to L2

Answer 1

Given that ,#7F (gC) ger)drc f(r)fen) dr Now fin) is an enen Funotion .. 0 firidr 0 2ferdr - 0 epneeut j fenj0 anea <9,7 (i) ,th Srgenydrtfny hon) dr Kt,a)t ,h) 1 -1

; CEIF 'ce) gjdn Sinee it satiefPies a the puctulaty Ofingen pra ouet. so that tt definag an innet produet.

Given thot Lo (r) = 1, h(A)= r 2 LoCre), Lur, 2ir) ARume thot ane Onthogonat Funetiou uer)d Lo Cr)t B LICn), dipeRpto wtr), Lotn)7 wer) LoCR) dr. 1 (()d+(> t LoCr)dn tPLi (r))dr othogona IxdnPxo an to Lotry wir) is not perpeudicul (L 11

Kwr), Le)wire) tn)dr wtr)ndr -1 -I -S(Acr)tpL)Ltr cr) dr Cra) aLo)drBLd 1 1 -l 0 duetu (Lo, 2 K4I, L270 0pthosf functiong wr) perpendicul a to r) LL