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If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$-norm (i.e, $p^\text{th}$-power-integrable functions), while other use the notation $L^p$ for the same purpose. If you are a functional analyst that distinguishes between $L^p$ and $L_p$, then could you please let me know what $L_p$ means (presumably, your definition of $L^p$ coincides with Wikipedia's definition)?

Now, I found another similar notation $\mathscr{L}_p$. What is $\mathscr{L}_p$? There seem to be $\mathscr{L}^p$, $\mathcal{L}^p$, and $\mathcal{L}_p$ too. However, I would expect that this is just due to people's using different fonts, i.e., $L^p=\mathscr{L}^p=\mathcal{L}^p$ and $L_p=\mathscr{L}_p=\mathcal{L}_p$.

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    $\begingroup$ I have seen $L_p$ being used as the space containing actual functions with finite $\Vert \cdot\Vert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is. $\endgroup$
    – Jan Bohr
    Commented Dec 11, 2018 at 21:20
  • $\begingroup$ Perhaps you're thinking of the distinction between $L_p$ spaces and $\mathscr L_p$ spaces. $\endgroup$
    – Robert Israel
    Commented Dec 11, 2018 at 21:27
  • $\begingroup$ @RobertIsrael What are $L_p$ and $\mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.) $\endgroup$
    – Batominovski
    Commented Dec 11, 2018 at 21:28
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    $\begingroup$ I learned it like this (from Dirk Werner personally :) )... $\mathscr L^p$ denotes the space of functions with finite $||\,\cdot\,||_p$-semi norm and $L^p$ is the space of equivalence classes $f \sim g$ if $f=g$ almost everywhere. The classes are such that $||\,\cdot\,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols. $\endgroup$
    – Nico
    Commented Dec 11, 2018 at 22:07

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The notation "some kind of L" with p "somwhere next to it" is so overused. It can mean:

  1. The space of measurable functions on some measure space with finite $p$ semi-norm. These objects are rarely used on their own. Usually they are a prelimiary steps for constructing Lebesgue spaces.

  2. The Lebesgue spaces. They are a quotient of the former spaces by subspaces of functions with zero $p$ semi-norm. In case of counting measre we use special notation: $\ell_p$.

  3. The Lindenstrauss-Pelczynski spaces. These space on a finite dimensional level level look just like finite dimensional $\ell_p$ spaces. See Absolutely summing operators in Lp-spaces and their applications J. Lindenstrauss; A. Pełczyński, Studia Mathematica (1968)

  4. The non-commutative Lebesgue space constructed for a given von Neumann algebra and a weight. This is a far reaching generalization, where you replace ordinary functions with bounded linear operators. See Non-Commutative Lp-Spaces G. Pisier, Quanhua Xu

  5. ...

This is not a complete list since there are so many ways to generalize classical Lebesgue spaces. Always check the definition used by your book or article.

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