# Definition A [[Map|function]] $g(x_1, x_2, \ldots, x_n)$ defined on an [[Open Set|open region]] $S$ of $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) is called *square-integrable* if the multiple integral $ \int_S [g(x_1, x_2, \ldots x_n)]^2 dx_1 dx_2 \ldots dx_2 $ exists. It is a theorem of complex analysis that any square-integrable function $g$ may be approximated by an [[Analytic Functions|analytic function]] $g'$ in such a way that the integral of $(g - g')$ over $S$ may be made as small as one wishes. # Vector Space To take a simpler example, consider all $\mathbb{F}$-valued ($\mathbb{F}$ is a [[field|field]], e.g., $\mathbb{R}$) functions $f(x)$ defined on $[a,b] \in \mathbb{R}$ that obey $ \int_{a}^{b} |f(x)|^2 \text{d}x <\infty $ These are the 1D square integrable functions and form the [[vector space]] $L^2([a,b])$. [[Vector]] addition and [[scalar]] multiplication are defined as: $ \begin{align} (f + g)(x) &= f(x) + g(x), \\ (cf)(x) &= cf(x) \end{align} $ and the zero element is $f(x) \equiv 0 \forall x \in [a,b]$. This is a vector space over the field $\mathbb{F}$. ## Applications in Physics the vector space $L^2(S)$ is the [[vector space]] of normalizable wavefunctions in quantum mechanics.