Orthogonal basis

Any orthogonal basis can be used to define a system of orthogonal coordinates ${\displaystyle V.}$ Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.

The concept of an orthogonal (but not of an orthonormal) basis is applicable to a vector space ${\displaystyle V}$ (over any field) equipped with a symmetric bilinear form ${\displaystyle \langle \cdot ,\cdot \rangle ,}$ where orthogonality of two vectors ${\displaystyle v}$ and ${\displaystyle w}$ means ${\displaystyle \langle v,w\rangle =0.}$ For an orthogonal basis ${\displaystyle \left\{e_{k}\right\}:}$

$${\displaystyle \langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}}$$

where ${\displaystyle q}$ is a quadratic form associated with ${\displaystyle \langle \cdot ,\cdot \rangle$ :} ${\displaystyle q(v)=\langle v,v\rangle }$ (in an inner product space ${\displaystyle q(v)=\|v\|^{2}.}$).

Hence for an orthogonal basis ${\displaystyle \left\{e_{k}\right\},}$

$${\displaystyle \langle v,w\rangle =\sum _{k}q(e_{k})v^{k}w^{k},}$$

where ${\displaystyle v_{k}}$ and ${\displaystyle w_{k}}$ are components of ${\displaystyle v}$ and ${\displaystyle w}$ in the basis.

• Basis (linear algebra) – Subset of a vector space that allows defining coordinates
• Orthonormal basis
• Orthonormal frame – Geometric structure that generalizes the Euclidean space
• Schauder basis
• Total set

• Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585, ISBN 978-0-387-95385-4
• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016.

• Weisstein, Eric W. "Orthogonal Basis". MathWorld.