Question

Problem 10. Let E be a measurable set and .ge L?(E). From the linearity of integration show that for any number : * Sp+21 ./+9+S4-+*20 From this and the quadratic formula derive directly the Cauchy - Schwarz Inequality.

Answer 1

© Poloblem -10;) Given that, het E be a measurable set and f, 9G-L² (E). [ Sep? +2187 $.9+ 8€ 92: SE (+p+9)' zo Sod.) het us consider Here t be any real number, we already know that integration is linear. SE 8² + 2+ SE P. G + SE 9² = SE (If +9) 2 20 we already know that (24 +9]2 is non-negalive function Poor all ti Now, at² + 2bt +LZO, VEER 0 Here, a = So 22 ; b = Stig ; C = ( 92 Therefore, the discrimant of the quadratic equation is co (26)2 - 4acco - b² Lac them (Se f.g2 < (+39.($90) Scanned with CamScanner

--}<1,9%) = 11417.11911 i. It is know as e Cauchy schwarz Inequality C Scanned with CamScanner