Square-Integrable Complex-Valued Function

functional-analysis

Definition

Square-Integrable Function

A square-integrable function is a complex-valued function whose squared magnitude has a finite integral over its domain.

For a function $f:[a,b]\to \mathbb{C}$, this means

$${\int }_a^b|f(x){|}^{2}dx<\infty .$$

Such functions are exactly the elements of L2 Space.

If $f$ is real-valued, the condition becomes

$${\int }_a^{b}f(x{)}^{2}dx<\infty .$$