# definition of the space of functions $L^\infty_{\text{loc}}(\mathbb{R}^n)$?

I couldn't find it on Google. I know that $L^1_{\text{loc}}(\mathbb{R}^n)$ is the space of locally integrable functions, but what about $L^\infty_{\text{loc}}(\mathbb{R}^n)$?

Thanks a lot!

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$f \in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^n)$ if, for every compact set $K \subset \mathbb{R}^n$, $f \in L^{\infty}(K)$. –  Christopher A. Wong Nov 21 '12 at 2:53
Thanks Christopher! –  Peter Valk Nov 21 '12 at 3:02

In general, $f\in L^p_{\text{loc}}(\mathbb{R}^n)$ if for every compact $K\subset\mathbb{R}^n$, $f\chi_K\in L^p(\mathbb{R}^n)$ where $\chi_K$ denotes the indicator function of $K$.
Note the minor edit - $f\vert_K$ didn't quite make sense in this context. –  icurays1 Nov 21 '12 at 3:03