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 {ek}kB of elements of H satisfying the conditions:

  1. Orthogonality: Every two different elements of B are orthogonal: ⟨ek, ej⟩= 0 for all k, j in B with kj.
  2. Normalization: Every element of the family has norm 1:||ek|| = 1 for all k in B.
  3. Completeness: The linear span of the family ek, kB, 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, ek⟩ = 0 for all kB and some vH 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 {fn : nZ} with fn(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.

Read more about this topic:  Hilbert Space

Other articles related to "orthonormal bases, orthonormal, base":

Hilbert Space - Orthonormal Bases - Separable Spaces
... A Hilbert space is separable if and only if it admits a countable orthonormal basis ... An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable ...
Orthogonal Group - Principal Homogeneous Space: Stiefel Manifold
... group O(n) is the Stiefel manifold Vn(Rn) of orthonormal bases (orthonormal n-frames) ... In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point given an orthogonal space, there is no natural choice of orthonormal basis ... The other Stiefel manifolds Vk(Rn) for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces any k-frame can be taken ...
Orthonormal Basis - As A Homogeneous Space
... The set of orthonormal bases for a space is a principal homogeneous space for the orthogonal group O(n), and is called the Stiefel manifold of orthonormal n-frames ... In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point given an orthogonal space, there is no natural choice of orthonormal basis, but once one ... The other Stiefel manifolds for of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces any k ...

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