# Schwarz inequality for multiple integrals: \$left[int_A f(x)g(x) dxright]^2 le int_A f(x)^2 dx cdot int_A g(x)^2 dx\$

If \$f,g:Ato mathbb{R}\$ are integrable, prove the Schwarz inequality

\$\$left[int_A f(x)g(x) dxright]^2 le int_A f(x)^2 dx cdot
int_A g(x)^2 dx\$\$

This is that type of question that isn’t easy to think about. I’ve found a solution for one dimension integrals here https://math.stackexchange.com/a/1089206/166180

But now I’m working on multiple integrals, so I’m not integrating over the real line, but in blocks. \$int_A f(x) dx\$ is defined as integrable when \$underline{int_A} f(x) dx = overline{int_A} f(x) dx\$, and these integrals are the sup and inf of those \$M_B\$ and \$m_b\$ things.

Does the proof I found apply for this case? It still makes sense for me when I switch \$int_a^b\$ by \$int_A\$, because the center of the proof is the equality \$(|f|+lambda|g|)^2 = |f|^2 + 2lambda |f||g| + lambda^2 |g|^2\$

## Answer

The integral of a non-negative, integrable function is non-negative. It follows that for any \$lambdainmathbb{R}\$ we have \$int_{A}left(f(x)-lambda g(x)right)^2,dx geq 0\$, so the discriminant of the quadratic polynomial

\$\$ p(lambda) = left(int_A g(x)^2,dxright)lambda^2 -2left(int_A f(x),g(x),dxright)^2 lambda + left(int_A f(x)^2,dxright)\$\$
is non-positive. Just write down the discriminant in terms of the coefficients and you are done.

Anyway, since in general \$L^1(A)notsubseteq L^2(A)\$, I believe that the correct assumptions should be about the square-integrability of \$f\$ and \$g\$. Or about \$f,gin L^1(A)\$, plus \$mu(A)<+infty\$.