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Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero:

$\left[\int_{\mathbb{R}^d}f(x)g(x)dx\right]^2+\left[\int_{\mathbb{R}^d}g(x)dx\right]^2>0$ ?


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up vote 1 down vote accepted

I don't think you can (if the space is over $\mathbb{R}$): Consider $f\in L^2$, then $f=f_1+f_2$ with $f_1\in L_0^2$ and $f_2\in (L_0^2)^{\perp}$ ($L_0^2$ is the space of $L^2$ functions with zero average). By Gram-Schmidt there is $g\in L_0^2$ such that $(f_1,g)_{L^2}=0$. But then $(f,g)_{L^2}=0$. So for this $g$ your LHS is zero.

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