Orthogonal set ($⟨u,v⟩=0(u\ne v)$)

An orthogonal set in an inner product space is a set whose distinct vectors have zero inner product. The zero vector may be included under this definition, but every nonzero member is linearly independent of the other nonzero members. An orthonormal set additionally requires every member to have norm one.

To prove independence, suppose ${\sum }_{j=1}^m{a}_j{v}_j=0$ for distinct nonzero orthogonal vectors. With the inner product linear in the first argument, pairing with $v_k$ gives $a_k\Vert v_k{\Vert }^2=0$. Positivity of the squared norm forces $a_k=0$ for every $k$.

The standard coordinate vectors in ${\ell }^2$ form an infinite orthonormal set. Their finite linear span contains only finite-support sequences, but its closure is the whole Hilbert space. Thus completeness of an orthonormal basis refers to closed span, not to every vector being a finite linear combination. A set can be orthogonal without spanning the space.