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I read a few things about $L^2$-Spaces and I am not at all sure whether I understand it right. So here are two question which I struggle with:

If a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is positive and bounded, is $f\in L^2((0,1)^n)$?

And are bounded functions in $ L^2((0,1)^n)$ also in $L^2(\mathbb{R}^n)$?

Laila
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    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Kavi Rama Murthy Mar 23 '21 at 11:37

1 Answers1

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For your first question : yes because constant function are $L^2((0,1)^d)$. For your second, no. Take for example $f(x)=x^2$ (with $d=1$).

joshua
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  • Thanks! But then what does $f$ is in $L^2((0,1)^d)$ tell me about the function? – Laila Mar 23 '21 at 11:48
  • that $\int_{(0,1)^d}f^2<\infty $. Also that $f\in L^1((0,1)^d)$ @Laila – joshua Mar 23 '21 at 11:49
  • Okay, I know the definition, but I think I don't really understand what it means. Does it describe other properties of the function in any way? – Laila Mar 23 '21 at 11:52