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Consider the space of functions $$f: \mathbb{R} \to \mathbb{R}$$ such that $$f(0) = 0$$ and $$\int_{-\infty}^{\infty} f'(x)^2 dx < \infty$$.

Does this sort of space have a common notation or name?

Note that this is not the same space as $L^2_1$, since I do not require the functions themselves to be $L^2$. For example the function $$f(x) = \tanh(x)$$ is not $L^2$ but is in the above space.

graviton
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  • Do you want $f$ to be continuous? What is $f’$? – blamethelag Dec 24 '22 at 18:00
  • You could define this space to be the completion of the smooth functions with compact support such that $f(0) = 0$ under the norm given by $|f|^2 = \int_{-\infty}^{\infty} f'(x)^2 dx$ – graviton Dec 25 '22 at 19:03
  • I don't know any notation for such a space. Maybe $\overline{\mathcal D _0(\mathbb R)}^{H^1}$ would do the job, where $\mathcal D _0 = \mathcal D \cap { f(0 ) =0 }$ is not conventional but cannot lead to confusion. – blamethelag Dec 26 '22 at 12:41

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