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The lebesgue space $L^{\infty}(\mathbb{R})$ means that all functions, such that:

$\big(\int_{-\infty}^{\infty}|f(x)|^{\infty}\big)^{1/\infty}<\infty\,\,?$

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    No, obviously not, did someone try to tell you this? It is not very hard to find the correct definition of $L^\infty$, it is basically the space of all bounded functions, up to a.e. equivalence2012-10-29

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No. What you have written is not well defined. It is an abuse of the $\infty$ notation. Check out "Walter Rudin: Real and Complex Analysis" for the proper definition and also an interesting problem in the Lp spaces chapter on how L infinity is a "limit" in Lp.

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As others have said, no. The reason for the NOTATION $L_\infty$ is that (under the right conditions) $ \|f\|_\infty = \lim_{p \to \infty} \|f\|_p $ You will find in the older books (say Banach's book) notations like $L_p$ used for $p<\infty$ but something else like $M$ instead of $L_\infty$.