Question

- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by <, >. Note: Feel free to use that continuous functions defined on closed bounded intervals have a maximum and a minimum value.

Answer 1

a defines an jener product in v ui cas 1. C, la Linearity : ut f.gh tv # <f+g, h a = a(x) + + g(x) hin) da sacx) h(x) • [f(n) + g(x)) da S ala) f(x) hen) dat l'ala) gin) h (2) de - cf, h7 at <g,h)a # For a scaler & and f, gtu. <44,97a - Saw (4 85 () nea) do a(a). d. f(n) h(a) da = a l'ault (2)h(2) dn O d < figna 2. symmetry, for , gev (f, 97 = a(a)t(n) g(x) da s'aca) g(n) fredda & (9f7a

3. & Positively for each Define ser. <5,172 - S'am fm) fou) dx - acx) [f(as) donant aca) >,0 x since, a is positive continuous ſh. so [f(x)] 20 11 tab and s aca) if (2) 72 da 7.0 d el Sol O 28,4720 fev 11,1 nm » s'aww(fons’de eo 4x since, and t (27]2 a is positive is also a ets fh non-negative ala)70 function 4 => a() [fea) Lo 5 (2) ² = 0 e. but a (2) >0. ae ta an f(2)=0 al 8x or f= 0 a.l. Also, Also, 40,07 = l'ae. oda = a (2) 0 ca.

ut da be the metric induced by inner product <,la da(x, y) = |, 11x-ylla | 23-9, X-9 , and let d be the netic induced by the inner product le d(x,y) = 11x-ylli - Ex-y, x-y> Consider şans is cauchy with the metric defined by <a N such that I a positive integer ie. For each exo & mn >N d. (xm, an) ce 12 dian, um 2 & ² inam) » (x₂-x, xn-ema ce s'aw tenXm) a) de ce since, a is a positive continuous, real valued function on a compact set toll, so it attains it's maximum and minimum , let Mo denotes maximum of 5 a (a) and mo denotes minimum of aca) so. 2 s' tcx.-R.)(as)’am corp

s On-ym)la do centro Xn- 2xn-xm, xn-'m Imo No Hence, senz is cauchy with the metric defined by nown Now Suppose Sang is metric defined by cauchy with for each exo i natural no. such that NEIN & m, n > N san- km) (2 7² de ce? Now, positive real valued Ho is the continuous maximum of function a continuous S'llanm) x 7 de ame? s'acullen-mX6m) on < MS 'reza-badew7 às < MER

» s'aws [p.--- you ’so che m by Oy or llen-Hollac The wym sans is cauchy with imetne defined by cra hence,

I hope it helps. Please feel free to revert back with further queries.