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**ragnar** Reading through Apostol's analysis text on $\displaystyle L^{2}(I)$ spaces, he claims that the inequality $\displaystyle |(f, g)| \leq ||f|| \cdot ||g||$ follows immediately from $\displaystyle \displaystyle \int_{I} \Bigl[ \int_{I} |f(x)g(y) - f(y)g(x)|^{2} dy \Bigr] dx \geq 0$. But all I can see is that $\displaystyle |(f, g)| = \Bigl| \displaystyle \int_{I} f(x)g(x) dx \Bigr| \leq \int_{I} |f(x)g(x)| dx $ and $\displaystyle \displaystyle ||f|| \cdot ||g|| = \Bigl( \bigl[ \int_{I} f^{2}(x) dx \bigr] \bigl[ \int_{I} g^{2}(x) dx \bigr] \Bigr)^{\frac{1}{2}}$, and I don't see how these three facts combine to prove the inequality we want. I've thought about squaring both and comparing them, but that doesn't seem like it's helping me.