Ph.D. Qualifying Exam – Spring 2004

Fifth problemProve that there exists an orthonormal basis B of L 2 ([0, 1]) such that∀f ∈ BLet S =∫ 10|f(x)| dx ∫ 1x < ∞ and f(x) dx x = 0Solution{f ∈ L 2 (0, 1) ∣ ∣∫ 1∀f ∈ S Tf =0∫ 10|f(x)| dx x < ∞ }0f(x) dxxand ∀n ∈ N g n =1 [1n ,1]Then S is dense in L 2 (0, 1). Indeed, let f be an L 2 function. For every positive integer n,the function f n = fg n is in S , andby dominated convergence.Define now‖f − f n ‖ 2 2 = ∫ 1n0Z = Ker Tf(x) dx −−−−→n→∞ 0and let’s show that Z is dense in S . For this, we will need the value of Tg n , which isnot a problem:∀n ∈ N Tg n =∫ 1Let f be any function in S . Then(T f − Tf )g n = 0Tg n1ndxx= ln n −−−−→n→∞ +∞which givesButand thereforef − TfTg ng n ∈ Z|Tf||Tg n | ‖g n‖ 2 = Tf (1 − 1 )−−−−→ln n n0n→∞(lim f − Tf )g n = fn→∞ Tg n8