Skip to content Skip to navigation


You are here: Home » Content » Orthogonal Bases


Recently Viewed

This feature requires Javascript to be enabled.

Orthogonal Bases

Module by: Mark A. Davenport. E-mail the author

Definition 1

A collection of vectors BB in an inner product space VV is called an orthogonal basis if

  1. span ( B ) = V span ( B ) = V
  2. vivjvivj (i.e., vi,vj=0vi,vj=0) ijij

If, in addition, the vectors are normalized under the induced norm, i.e., vi=1ivi=1i , then we call VV an orthonormal basis (or “orthobasis”). If VV is infinite dimensional, we need to be a bit more careful with 1. Specifically, we really only need the closure of span (B) span (B) to equal VV. In this case any xVxV can be written as

x = i = 1 c i v i x = i = 1 c i v i

for some sequence of coefficients {ci}i=1.{ci}i=1.

(This last point is a technical one since the span is typically defined as the set of linear combinations of a finite number of vectors. See Young Ch 3 and 4 for the details. This won't affect too much so we will gloss over the details.)

Example 1

  • V=R2V=R2, standard basis

Example 2

  • Suppose V={ piecewise constant functions on [0,14),[14,12),[12,34),[34,1]}V={ piecewise constant functions on [0,14),[14,12),[12,34),[34,1]}. An example of such a function is illustrated below.
    Figure 1
    An example of a piecewise constant function.
    Consider the set
    Figure 2
    Illustrations of the Haar basis functions. v_1 is the all constant function, i.e., it is 1 on [0,1].
    Illustrations of the Haar basis functions. v_2 is 1 on [0,0.5) and -1 on [0.5,1].
    Illustrations of the Haar basis functions. v_3 is sqrt(2) on [0,0.25), -sqrt(2) on [0.25,0.5), and 0 on [0.5,1].
    Illustrations of the Haar basis functions. v_4 is 0 on [0,0.5), sqrt(2) on [0.5,0.75), and -sqrt(2) on [0.75,1].
    The vectors {v1,v2,v3,v4}{v1,v2,v3,v4} form an orthobasis for VV.
  • Suppose V=L2[-π,π]V=L2[-π,π]. B={12πejkt}k=-B={12πejkt}k=-, i.e, the Fourier series basis vectors, form an orthobasis for VV. To verify the orthogonality of the vectors, note that:
    See Young for proof that the closure of BB is L2[-π,π]L2[-π,π], i.e., the fact that anyfL2[-π,π]fL2[-π,π] has a Fourier series representation.

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens


A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks