I need to proove the Minkowski's inequality for integrals. I am taking a course in analysis. [ int(f+g)^2 ] ^(1/2) =< [int(f^2)]^(1/2) + [int(g^2)]^(1/2) now we are given that both f and g are Riemann integrable on the interval. So by the properties of Riemann integrals, so is f^2,g^2 and fg. We are also given a hint to expand the integral on the left and then use the Cauchy-Bunyakovsky-Schwarz inequality (now this i've already prooved in a previous exercice using the discriminant). I was trying to expand the left side but i don't know what to do with the squared root, moreover i was trying to expand regardless the squared root and then at the end take a squared root but it still hasn't worked.. I need help =) Thanks, Joe
form what you wrote i assume you have an inner product given by <f,g> = int fg dx and the induced norm |f| = <f,f>^½ so the Cauchy-Schwarz inequality is |<f,g>| <= |f||g| from what you get <f,g> <= |f||g| so starting with |f+g| = <f+g.f+g> = (int (f+g)^2)^½ = (int (f^2+g^2+2fg)^½ = (int f^2 + int g^2 + int 2fg)^½ = (<f,f>+<g,g>+2<f,g>)^½ = (|f|^2+|g|^2+2<f,g>)^½ by cauchy-schwarz <= (|f|^2+|g|^2+2|f||g|)^½ = [(|f|+|g|)^2]^½ = |f|+|g| qed.