= Orthogonal set
{title2=$\langle u,v\rangle=0\quad(u\ne v)$}
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 $\sum_{j=1}^ma_jv_j=0$ for distinct nonzero orthogonal vectors. With the inner product linear in the first argument, pairing with $v_k$ gives $a_k\|v_k\|^2=0$. 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.
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