## Minkowski's inequality!

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
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 form what you wrote i assume you have an inner product given by = int fg dx and the induced norm |f| = ^½ so the Cauchy-Schwarz inequality is || <= |f||g| from what you get <= |f||g| so starting with |f+g| = = (int (f+g)^2)^½ = (int (f^2+g^2+2fg)^½ = (int f^2 + int g^2 + int 2fg)^½ = (++2)^½ = (|f|^2+|g|^2+2)^½ by cauchy-schwarz <= (|f|^2+|g|^2+2|f||g|)^½ = [(|f|+|g|)^2]^½ = |f|+|g| qed.
 i forgot a ^½ int the line |f+g| = it should be |f+g| =
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