Hilbert Space - Orthonormal Bases

Orthonormal Bases

The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an orthonormal basis is a family {e_k}_{k ∈ B} of elements of H satisfying the conditions:

1. Orthogonality: Every two different elements of B are orthogonal: ⟨e_k, e_j⟩= 0 for all k, j in B with k ≠ j.
2. Normalization: Every element of the family has norm 1:||e_k|| = 1 for all k in B.
3. Completeness: The linear span of the family e_k, k ∈ B, is dense in H.

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if B is countable). Such a system is always linearly independent. Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:

if ⟨v, e_k⟩ = 0 for all k ∈ B and some v ∈ H then v = 0.

This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if S is any orthonormal set and v is orthogonal to S, then v is orthogonal to the closure of the linear span of S, which is the whole space.

Examples of orthonormal bases include:

• the set {(1,0,0), (0,1,0), (0,0,1)} forms an orthonormal basis of R3 with the dot product;
• the sequence {f_n : n ∈ Z} with f_n(x) = exp(2πinx) forms an orthonormal basis of the complex space L2;

In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.