Inner product of two Schwartz fucntions.

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Say I have two Schwartz fucntions $f(x)$ and $g(x)$.

Does $\int f(x)g(x)dx \approx 0$ imply that $f(x)$ $g(x)$ have virtually non-overlapping support ?

Thank you for your time.

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No, just take $f$ even and $g$ odd.