Square Integrable Functions, Norms, and the Trial Method

by Gaurav Tiwari October 3, 2026

This is Part II of the Integral Equations series. If you haven’t read Part I on definitions and types, start there.

Here we cover the function spaces that integral equation theory lives in: square integrable functions ($ \mathfrak{L}_2 $ spaces), inner products, norms, and the key inequalities. Then we solve an integral equation using the trial method, step by step.

This article is adapted from my free 25-page book Elements of Integral Equations, which works through 26 examples with every computation shown.

Square Integrable Functions ($ \mathfrak{L}_2 $ Functions)

A function $ y(x) $ is said to be square integrable (or $ \mathfrak{L}_2 $) on the interval $ (a,b) $ if

$$ \int_a^b |y(x)|^2 \, dx < \infty $$

or equivalently,

$$ \int_a^b y(x) \bar{y}(x) \, dx < \infty $$

Such a function is also called a regular function. The space of all square integrable functions on $ (a,b) $ is denoted $ L^2(a,b) $ and forms a Hilbert space, the natural setting for integral equation theory.

The kernel $ K(x,t) $, a function of two variables, is an $ \mathfrak{L}_2 $-function if at least one of the following holds:

$ \int_{x=a}^b \int_{t=a}^b |K(x,t)|^2 \, dx \, dt < \infty $
$ \int_{t=a}^b |K(x,t)|^2 \, dx < \infty $
$ \int_{x=a}^b |K(x,t)|^2 \, dt < \infty $

Inner Product of $ \mathfrak{L}_2 $ Functions

The inner product (or scalar product) $ (\phi, \psi) $ of two complex $ \mathfrak{L}_2 $ functions $ \phi $ and $ \psi $ of a real variable $ x $ on $ [a, b] $ is defined as

$$ (\phi, \psi) = \int_a^b \phi(x) \bar{\psi}(x) \, dx $$

where $ \bar{\psi}(x) $ is the complex conjugate of $ \psi(x) $.

When $ (\phi, \psi) = 0 $, the functions $ \phi $ and $ \psi $ are called orthogonal to each other. Orthogonality is central to the spectral theory of integral equations, where eigenfunctions of symmetric kernels form orthogonal systems.

Norm of a Function

The norm of a complex function $ y(x) $ on $ [a,b] $ is

$$ \| y(x) \| = \sqrt{\int_a^b |y(x)|^2 \, dx} $$

The norm measures the “size” of a function in $ L^2 $. A function with $ \|y\| = 0 $ is the zero function (almost everywhere). Two fundamental inequalities govern how norms interact:

Schwarz’s Inequality: $ |(\phi, \psi)| \le \|\phi\| \cdot \|\psi\| $

Triangle Inequality (Minkowski): $ \| \phi + \psi \| \le \|\phi\| + \|\psi\| $

These inequalities are not just theoretical niceties. They’re the tools you need to prove convergence of iterative methods for solving integral equations, which we’ll cover in later parts of this series.

Solution of Integral Equations by the Trial Method

The trial method is the simplest approach to solving an integral equation: you’re given (or guess) a candidate solution and verify it satisfies the equation. It’s not a general-purpose solver, but it builds the intuition you need before tackling iterative and resolvent kernel methods.

Worked Example: Volterra Equation of the Second Kind

Problem. Show that $ y(x) = (1+x^2)^{-3/2} $ is a solution of

$$ y(x) = \frac{1}{1+x^2} – \int_0^x \frac{t}{1+x^2} \, y(t) \, dt \quad \ldots (1) $$

This is a Volterra equation of the second kind with lower limit $ a = 0 $ and upper limit the variable $ x $.

Solution. We have $ y(x) = (1+x^2)^{-3/2} $, so $ y(t) = (1+t^2)^{-3/2} $ by replacing $ x $ with $ t $.

Substituting into the RHS of (1):

$$ \text{RHS} = \frac{1}{1+x^2} – \int_0^x \frac{t}{1+x^2} (1+t^2)^{-3/2} \, dt $$

Since $ \frac{1}{1+x^2} $ is independent of $ t $, it factors out:

$$ = \frac{1}{1+x^2} – \frac{1}{1+x^2} \int_0^x \frac{t}{(1+t^2)^{3/2}} \, dt $$

Evaluating the integral (substituting $ u = 1 + t^2 $):

$$ \int_0^x \frac{t}{(1+t^2)^{3/2}} \, dt = 1 – \frac{1}{\sqrt{1+x^2}} $$

Substituting back:

$$ \text{RHS} = \frac{1}{1+x^2} + \frac{1}{1+x^2}\left(\frac{1}{\sqrt{1+x^2}} – 1\right) = \frac{1}{(1+x^2)^{3/2}} = y(x) $$

Hence $ y(x) = (1+x^2)^{-3/2} $ is a solution of (1). $ \square $

Continue the Series

This article is Part II. The complete integral equations series:

Part I: Definitions and Types of Integral Equations
Part II: Square Integrable Functions, Norms, and Trial Method (you are here)
Part III: Converting Differential Equations into Integral Equations
Part IV: Converting Integral Equations into Differential Equations

This series is adapted from Elements of Integral Equations, my free 25-page introduction. When you’re ready for iterative methods, resolvent kernels, Fredholm theory and numerical methods, with 321 exercises to work through, my book Integral Equations covers them.

Integral Equations: An Interactive Course in Theory, Methods, and Computation

24 chapters
321 exercises
60 figures
Kindle + paperback

24 chapters in 6 parts, from reading an equation to defending a solution
321 exercises, each with a hint or answer at the back
Fredholm and Volterra equations, separable kernels, the Neumann series and resolvents
Initial and boundary value problems in integral form, with Green's functions
The Fredholm alternative, symmetric kernels and spectral methods
Nyström, collocation, Galerkin and Tikhonov regularization for noisy data
6 benchmark problems revisited with several methods
Prerequisites: calculus, basic ODEs and linear algebra

$12.99

Kindle edition. Paperback $29.99.

Price checked October 3, 2026

A first course in integral equations built around choosing a method, solving the problem, and checking what the answer actually establishes.

Get the Kindle Edition Paperback on Amazon Kindle on Amazon.in

Free PDF with worked solutions to all 321 exercises

Square Integrable Functions, Norms, and the Trial Method

Square Integrable Functions (\( \mathfrak{L}_2 \) Functions)

Inner Product of \( \mathfrak{L}_2 \) Functions

Norm of a Function

Solution of Integral Equations by the Trial Method

Worked Example: Volterra Equation of the Second Kind

Continue the Series

Further Reading