If $f \in L((0,1)^n)$ bounded does follow that $f$ is in $L(\mathbb{R}^n)$?

Question

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 $L^2(\mathbb{R}^n)$?

Answer 1

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 $d=1$).