Question

Problem 2. We define <= {ue L’(0,1) : ſ'u(e)dt = 0} Firstly, prove that L is a closed subspace of L²(0,1). Moreover, we denote by an opera- tor PL Pr(u) = v= ['us)ds, u € 1 (0,1). Then, prove that P, is a projection operator from L? (0,1) onto L.

Answer 1

Solution: L- Suecia Suci We have to prove I ba Let SEL (04) be a binit point of L closed retspore of trad) such that en uniformly of ADHD f} care , ) Rock that fn E L th ft L, which will imply Lis cloled as ISL I f(t) dt J Lim, tult) de we show Now so as fuel, stult) at PL to lineal. Now, we have the of erator PL given by PL(u) = u sureds, ut 2 (61) We first show How Pl(utco = 4+ Sudstev. uteo- sure) rods , cla scalar on the integra seus aute SO PL es linear. ht ucs ds = 1 Now, RCPW)=P(ul) su-goods -ul surends + ls as u-l la l=u-leu-funde Suds- e sds PCWEL PL to a projection operator onto L. (proved) & P2 = PL So, PL is a projection of eroter. Now, S' Pruds SO