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Mathematics for Physics

236 Quadratic Integrability

A property of a function under which the integral of the square of its absolute value is finite that is used to say the function has finite size under a square integral.

definition [d] (Quadratic Integrability = Square Integrability) A property of a function \(f\): the integral of \(|f|^{2}\) is finite.

where

  • \(f\) is a function on a domain of integration.
  • \(|f|^{2}\) is the square of the absolute value of \(f\).
  • the scalar field of values of \(f\) may be \(\mathbb{R}\).
  • the scalar field of values of \(f\) may be \(\mathbb{C}\).

Note:

  • this is the integrability condition that defines the classical space \(L^{2}\).

236.1 Elementary Example

236.1.1 Simple

Quadratic integrability means \(\int |f|^{2}\) is finite. On three sample points, sum the squares of absolute values.

\[ A = \{ 1,\ 2,\ 3 \} \]

\[ f(1) = 1,\quad f(2) = -2,\quad f(3) = 1 \]

\[ \sum_{a \in A} |f(a)|^{2} = 6 < \infty \]

where

  • \(|f|^{2}\) is the square of the absolute value of \(f\).

236.1.2 General

On an interval, square integrability is finiteness of \(\int |f|^{2}\).

\[ f(x) = e^{-x^{2}}\ \text{on }\mathbb{R} \]

\[ \int_{-\infty}^{\infty} |f(x)|^{2}\, dx < \infty \]

where

  • this is the membership condition for the classical space \(L^{2}\).

236.2 References

  1. Pietsch, A. History of Banach Spaces and Linear Operators. Birkhäuser, 2007. — square integrability as finiteness of the integral of \(|f|^{2}\); Riesz.