An orthogonal basis of a real or complex inner product space 
,
in the algebraic sense, is a vector basis 
whose distinct vectors
are orthogonal. Thus the vector
space span of the vectors is 
, and
 |
where 
and 
is the Kronecker delta. In particular, every basis
vector is nonzero. If all
, the basis is an orthonormal
basis. More generally, dividing each 
by its norm produces an orthonormal
basis (Axler 2026, pp. 237 and 243-244).
In a Hilbert space, the term is also used for a family of nonzero, pairwise orthogonal vectors whose vector space span is dense. Normalizing such a complete orthogonal system produces a Hilbert basis; the completeness requirement uses the set closure of the vector space span rather than only finite linear combinations (Axler 2026, pp. 242-244).
