In mathematics, a square-integrable function, also called a quadratically integrable function or function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows. One may also speak of quadratic integrability over bounded intervals such as for. An equivalent definition is to say that the square of the function itself is Lebesgue integrable. For this to be true, the integrals of the positive and negativeportions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions form the Lp space with. Among the Lp spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the Lp spaces are complete under their respective p-norms. Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.
Properties
The square integrable functions form an inner product space with inner product given by where
is the set over which one integrates—in the first definition, is ; in the second, is.
Since, square integrability is the same as saying 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 spacesconvergeif 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 and many times abbreviated as. Note that 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 specify the inner product space. The space of square integrable functions is the Lp space in which.
Examples
, defined on, is in L2 for but not for.
Bounded functions, defined on . These functions are also in Lp, for any value of p.
, defined on.
Counterexamples
, defined on , where the value of f is arbitrary. Furthermore, this function is not in Lp for any value of p in.
