# Notations in Functional Analysis: $L^p$, $L_p$, $\mathscr{L}^p$, $\mathscr{L}_p$, $\mathcal{L}^p$, and $\mathcal{L}_p$

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 $$L_p=\mathscr{L}_p=\mathcal{L}_p$$.

• 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. Commented Dec 11, 2018 at 21:20
• Perhaps you're thinking of the distinction between $L_p$ spaces and $\mathscr L_p$ spaces. Commented Dec 11, 2018 at 21:27
• @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.) Commented Dec 11, 2018 at 21:28
• 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.
– Nico
Commented Dec 11, 2018 at 22:07

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.