# $u \in L^2(R^n)$ what does this mean?

$u \in L^2(R^n)$

I am guessing that $L^2(R^n)$ means the $L^2$ norm over an n-dimensional vector. The context is an energy minimization function : total variation–based model of Rudin, Osher, and Fatemi (ROF)

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I am not sure if I have given enough information about the context to expect a response. I can provide more, if needed –  AnkurVijay Jun 3 '11 at 18:15
I would guess that $L^2(\mathbb{R}^n)$ means the space of square-integrable functions with domain $\mathbb{R}^n$. –  Arturo Magidin Jun 3 '11 at 18:16
It would be most helpful if you provide more context, yes. –  Arturo Magidin Jun 3 '11 at 18:16

## 1 Answer

$L^2(\mathbb{R}^n)$ is the space of all measurable functions $f\colon \mathbb{R}^n \to \mathbb{R}$ (or possibly $f\colon \mathbb{R}^n \to \mathbb{C}$) such that $$\int_{\mathbb{R}^n} |f|^2 \;<\; \infty\text{,}$$ where the integral is a Lebesgue integral. (The square root of this integral is the 2-norm of $f$.)

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In some cases, $L^2 (\mathbb{R}^n)$ denotes the above space after identifying two functions if they differ almost everywhere (thus its element are in fact equivalence classes of functions). In most cases the distinction doesn't matter but in some it does. –  Mark Jun 3 '11 at 18:23
Mark is of course correct. The issue is that the 2-norm on the space I have defined is only a seminorm, with any function that is zero almost everywhere having norm 0. By identifying almost everywhere equal functions, the 2-norm becomes an actual norm on the quotient. –  Jim Belk Jun 3 '11 at 18:38
I dont understand why are we considering equivalence classes of functions and what is meant by "almost everywhere". –  AnkurVijay Jun 3 '11 at 18:48
IF you don't know about "almost everywhere" and such things, then probably that paper (or book) is not for you. –  GEdgar Jun 3 '11 at 18:54
@GEdgar yes i am finding it particularly difficult to go through the paper, but it is essential that i understand it. –  AnkurVijay Jun 3 '11 at 18:58