What does the curly x symbol "$\mathcal{X}$" mean in "$\;x_i\in\mathcal{X}\;$"?

Question

If you need more context, this equation is on the second page of this document. My guess is it just means "whatever value $x$ can have" or "all values of $x$", but I just wanted to check.

Answer 1

Capital letters such as $\mathcal{X}$ (calligraphy "X") in the picture provided can be used to denote a set; in this case, it is. However, this is not always the case as $f \subseteq \mathbb{R} \times \mathbb{R}$ (lower case script "f") is sometimes used to denote a function on $\mathbb{R}$. You may also see capital letters used to denote an element of a set such as in the definition of the limit of a sequence for example. It just depends on context and the author of the literature.