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The only one I am familiar with is $\mathcal{L}$ for Laplace transforms, but I have now encountered other symbols such as $\mathcal{F}$ and $\mathcal{M}$. I'm curious: does each calligraphic letter have a specific use case, or are they just alternatives to the standard $f$ and $F$ types (and other letters)?

Kalcifer
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    You can define anything. In some complex-analysis papers I saw how the authour named contour-paths $\mathcal{L}_j$. The three you stated may indicate Integral-Transformations: Laplace-T, Fourier-T and Mellin-T. But there are many other examples where one uses caligraphic letters, for example the Lagrangian. I've seen P.V. written as $\mathcal{P}$, and $\mathcal{D}$ for extravagant derivatives. – vitamin d Feb 23 '21 at 02:33
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    Notation means whatever the author wants it to mean. In the most general sense, yes they are just additional options that can be used in any setting. That said, you will commonly find mathcal used for the name of measure spaces or sigma algebras, for families of sets, and many others. See also. – JMoravitz Feb 23 '21 at 02:34
  • Math does not really have defined notational conventions (although it has very widely used notation). I think many people like to use them when you want to use capital letters but are already using those for other purposes, and you don't want to go into a completely different alphabet (e.g. Greek, Russian, or Hebrew, all commonly used for additional "letter supply" in math mostly typeset in Roman letters) – leslie townes Feb 23 '21 at 02:35
  • While some notations are popular I think it's fair to say that nothing is standard. Even symbols like $\pi$ which have a usual meaning attached to them that almost everyone including laymen are aware of will be repurposed by authors for their own ends. Personally I like calligraphic letters to represent topological spaces and collections of sets. – CyclotomicField Feb 23 '21 at 02:45

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So far I've seen:

$\mathcal{A}, \mathcal{B}$ for bases in linear algebra or function classes

$\mathcal{C}, \mathcal{D}$ for integration domains in the complex plane

$\mathcal{E}$ looks like a "capital epsilon ($\epsilon$)" so it might be used together with other capital letters (e.g. matrices) like $f(A+\mathcal{E})$

$\mathcal{F}, \mathcal{G}, \mathcal{H}$ for function classes

$\mathcal{I}, \mathcal{J}, \mathcal{K}$ for index sets (often uncountable)

$\mathcal{L}$ for linear functionals (linear map that takes in a function)

$\mathcal{M}$ (not sure)

$\mathcal{N}$ for normal distribution

$\mathcal{O}$ for big-O notation

$\mathcal{P}$ for power set

$\mathcal{Q}$ (not sure)

$\mathcal{R}$ for Rademacher complexity, or response function when solving DEs using Fourier transform

$\mathcal{S}, \mathcal{T}, \mathcal{U}, \mathcal{V}$ transformations (often linear)

$\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ domains of random variables or random samples

Benjamin Wang
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