# How to prove a function is square integrable?

## Question:

How to prove a function is square integrable?

## Definite Integral:

Let us consider a real-valued function of one variable {eq}f(x) {/eq} that is defined over the real axis {eq}(-\infty, \infty). {/eq}

The definite integral of the function is expressed by the following mathematical expression

{eq}\displaystyle \int_{-\infty}^{\infty} f(x) \; dx. {/eq}

## Answer and Explanation:

We are given a real-valued function of one variable {eq}y=f(x) {/eq} defined on the interval {eq}(-\infty, \infty). {/eq}

The function is said to be square-integrable if the definite integral of its squared absolute value is finite, i.e.

{eq}\displaystyle \int_{-\infty}^{\infty} |f(x)|^2 \; dx < \infty. {/eq}

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Integral Calculus: Definition & Applications

from JEE (Main): Study Guide & Test Prep

Chapter 12 / Lesson 2
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