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?
Best Answer
The space $L^2(\mathbb{R}^3)$ is indeed the vector space of all square-integrable functions from $\mathbb{R}^3$ into $\mathbb R$. Here, integrable means Lebesgue-integrable. This space has a natural norm: $\|f\|_2=\sqrt{\int_{\mathbb{R}^3}f^2}$.
More generally, if $p\geqslant1$ you have the space $L^p(\mathbb{R}^n)$ of all functions $f\colon\mathbb{R}^n\longrightarrow\mathbb R$ such that $|f|^p$ is Lebesgue-integrable. The natural norm here is $\|f\|_p=\left(\int_{\mathbb{R}^n}|f|^p\right)^{1/p}$.