Meaning of the notation $L^2(\mathbb{R}^3)$ and its generalization

Question

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?

Answer 1

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 $\|f\|_p=\left(\int_{\mathbb{R}^n}|f|^p\right)^{1/p}$.

Answer 2

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 $X$.

Answer 3

In measure theory, $L^p(\Omega)$ can be defined for various spaces and various measures. However, $p\geq 1$ is always necessary to make sure the operation

$$ \|f\|_{L^p}= \sqrt[p]{\int_{\Omega}|f(x)|^p\ \mathrm{d}\mu(x)} $$

is actually a norm. Some cases you might be interested in:

• $\Omega$ is some simple subset of $\mathbb{R}^p$ (a hypercube for example) and $\mu$ is the (usual) lebesgue measure
• $\Omega$ as above and $\mu$ is a weighted lebesgue measure. That is, roughly speaking, $\mathrm{d}\mu(x) = g(x)\mathrm{d}x$ for some positive weight function $g$. This yields (ignoring differences between rieman integrals and lebesgue integrals)

$$\|f\|_{L^p}^p=\int_\Omega |f(x)|^p\mathrm{d}\mu(x) = \int_\Omega |f(x)|^pg(x)\mathrm{d}x$$

Answer 4

One technicality that might be good to know but seldom needs to be considered:

The elements in $L^p(X)$ for some measure space $X$ are actually equivalence classes of functions.

If two functions $f$ and $g$ differ on a null set (a set of length/area/volume $0$, e.g. at a few points) then they cannot be differed by integration. That would make $\|f-g\| = 0$ although $f \neq g$, and $\|\cdot\|$ would not be a norm. Therefore functions differing only on null sets are considered equal (belonging to the same equivalence class), and $L^p(X)$ is then technically a set of equivalence classes.

Answer 5

The concept of Bochner integration allows one to define the spaces $L^p(X;Y)$ for $X,Y$ Banach spaces. They are like the normal $L^p$ spaces but take values in $Y$.