Square-integrable

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if

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

then ƒ is square integrable on the real line (-\infty,+\infty). One may also speak of quadratic integrability over bounded intervals such as [0, 1].[1]

Properties

The square integrable functions form an inner product space whose inner product is given by

\langle f, g \rangle = \int_A \overline{f(x)}g(x)\, dx

where

  • f and g are square integrable functions,
  • f(x) is the complex conjugate of f,
  • A is the set over which one integrates—in the first example above, A is (-\infty,+\infty); in the second, A is [0, 1].

Since |a|2 = a, square integrability is the same as saying

\langle f, f \rangle < \infty. \,

It can be shown that square integrable functions form a complete metric space under the metric induced by the inner product defined above. A complete metric space is also called a Cauchy space, because sequences in such metric spaces converge if and only if they are Cauchy. A space which is complete under the metric induced by a norm is a Banach space. Therefore the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product. As we have the additional property of the inner product, this is specifically a Hilbert space, because the space is complete under the metric induced by the inner product.

This inner product space is conventionally denoted by \left(L_2, \langle\cdot, \cdot\rangle_2\right) and many times abbreviated as L_2. Note that L_2 denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation. The set, together with the specific inner product \langle\cdot, \cdot\rangle_2 specify the inner product space.

The space of square integrable functions is the Lp space in which p = 2.

References

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