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: <math>\langle f, g\rangle_{L^2} = \int_{-\infty}^\infty f(x) \cdot \overline{g(x)} \, dx.</math>
Since this is a closed subspace of <math>L^2(\mathbb R)</math>, it is a HIlbert space. Moreover, the elements of <math>H</math> are smooth functions on <math>\mathbb R</math> that tend to zero at infinity, essentially by the [[Riemann-Lebesgue lemma]]. In fact, the elements of <math>H</math> are the restrictions to <math>\mathbb R</math> of entire [[holomorphic function]]s, by the [[
From the [[Fourier inversion theorem]], we have
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