Understanding the Orthonormal Basis

The world of mathematics can be very intimidating for many people. In fact, you could even say that only a select few truly enjoy math and the process of solving equations. There are countless terms, systems, and methodologies involved in math, which can double the difficulty level. If you’re studying a university course that focuses on mathematics or you simply enjoy it learning about it, then perhaps you’ve come across the term orthonormal basis. It can be a difficult concept to wrap your head around, which is why in this article, we’ll discuss and explain what orthonormal basis is in the simplest of terms so that anyone can understand.

What Is an Orthonormal Basis?

An orthonormal basis for an inner product space V (a vector space with an inner product) is essentially a set of dense elements in the space and contains elements that are orthogonal to each other. Let’s take the Euclidean space’s standard basis (Rn) as an example. This spaced is considered an orthonormal basis, and the relevant inner product is known as the dot product of vectors. If you take the image of the standard basis and put it under a rotation or reflection, it would still be orthonormal. As such, all the orthonormal bases for Rn arises in the same way.

If you’re looking at it from a more general perspective within an inner product space V, you can use an orthonormal basis to define normalized orthogonal coordinates within V. In any case, using the orthonormal basis is necessary for both finite-dimensional spaces as all as infinite-dimensional ones.

In Relation to Hamel Bases

It’s worth noting that if you’re dealing with an infinite-dimensional space, the term orthonormal basis wouldn’t be the same as the one in linear algebra. That’s why to tell the two apart, the latter bases are otherwise known as Hamel bases. When dealing with inner product spaces, you won’t really make use of Hamel bases. On the other hand, orthonormal bases in this circumstance are fundamental, so it’s necessary to distinguish the two.

Some Examples

  • If you take a set of vectors, namely the standard basis of {e1 = (1, 0, 0)e2 = (0, 1, 0)e3 = (0, 0, 1)}, it forms an orthonormal basis of R3.
  • The set {fn : n ∈ Z} with fn(x) = exp(2πinx) creates an orthonormal basis of L2([0,1]), otherwise known as the complex space. If you’re studying the Fourier series, in particular, this information will come in handy.
  • The set {eb : b ∈ B} with eb(c) = 1 if b=c and 0 becomes an orthonormal basis of l2(B).

Simple Formula

Here’s a simple formula you may want to check out. If B is an orthogonal basis of H, then you can write every element x of H as the following:

{\displaystyle x=\sum _{b\in B}{\frac {\langle b,x\rangle }{\lVert b\rVert ^{2}}}b.}

On the other hand, if B is orthonormal instead of orthogonal, then the formula above would have to be simplified into:

{\displaystyle x=\sum _{b\in B}\langle b,x\rangle b}


This is only an introduction to orthonormal bases, and the concept can definitely become more complex the more you deep-dive into it and learn more. However, we’ll end it here for now. Hopefully, you have a better idea of what an orthonormal basis is and that this information will help you in class or any math-related endeavor you may have.