Meaning of the notation $L^2(\mathbb{R}^3)$ and its generalization I'm not sure whether this question really belongs to this website. In quantum physics texts, and physics stackechange website, I have often seen the notation $L^2(\mathbb{R}^3)$. My glossary of mathematical notations are limited. What does this symbol precisely mean?
I think $L^2$ is a vector space of square-integrable functions in three-dimensional space $\mathbb{R}^3$. Is there anything more to it? What are some generalization of this notation?

That's all it is: the space of square integrable functions on $\mathbb R^3$. In set builder notation, $$L^2(\mathbb R^3) = \left\{f: \mathbb R^3 \to \mathbb C : \int_{\mathbb R^3} \lvert f(x) \rvert^2 dx < \infty \right\}.$$ This space is a complete inner-product space (i.e. a Hilbert space) with inner product $$\langle f,g \rangle = \int_{\mathbb R^3} f(x) \overline{g(x)} dx, \,\,\,\,\, f,g \in \mathbb R^3.$$ You  can read about this space (and the related $L^p(\mathbb R^3)$ spaces for $p \ge 1$) here: https://en.wikipedia.org/wiki/Lp_space. You can also change the underlying space from $\mathbb R^3$ to any measure space $X$.
