$u \in L^2(R^n)$
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 03 '11 at 18:15
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I would guess that $L^2(\mathbb{R}^n)$ means the space of square-integrable functions with domain $\mathbb{R}^n$. – Arturo Magidin Jun 03 '11 at 18:16
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It would be most helpful if you provide more context, yes. – Arturo Magidin Jun 03 '11 at 18:16
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$L^2(\mathbb{R}^n)$ 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 $f$.)
Jim Belk
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1In 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 03 '11 at 18:23
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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 03 '11 at 18:38
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I dont understand why are we considering equivalence classes of functions and what is meant by "almost everywhere". – AnkurVijay Jun 03 '11 at 18:48
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IF you don't know about "almost everywhere" and such things, then probably that paper (or book) is not for you. – GEdgar Jun 03 '11 at 18:54
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@GEdgar yes i am finding it particularly difficult to go through the paper, but it is essential that i understand it. – AnkurVijay Jun 03 '11 at 18:58
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@AnkurVj, if the paper doesn't use the phrase "almost everywhere" (which you can look up on Wikipedia), it's possible you don't need to worry about it. However, it's also possible that you will need to learn more about measure theory and Lebesgue integration before you can read the paper successfully. – Jim Belk Jun 03 '11 at 20:34
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@Jim Further in the paper i found the usage and brief explanation of the term "almost everywhere". Not having the proper mathematical background i couldn't guess "almost everywhere" refers to some precise mathematical notion and hence didn't think that looking up on wikipedia would help. I agree that i need to learn more about measure theory. – AnkurVijay Jun 03 '11 at 20:47
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The best place (in my opinion) to learn about basic measure theory is Walter Rudin's book "Real and Complex Analysis". The first three chapters furnish a decent background in the subject. However, once you read the first three chapters, you will not be able to stop! – Amitesh Datta Jun 03 '11 at 23:42