Orthonormal basis - Wikipedia

Abstraction: Basis of mutually orthogonal unit vectors spanning inner product space

Key points:

• An orthonormal basis for a finite-dimensional inner product space is a set of vectors that are pairwise orthogonal and each of unit norm; every vector can be expressed as a simple dot-product linear combination.
• Any finite-dimensional inner product space has an orthonormal basis, obtainable from an arbitrary basis via the Gram–Schmidt process.
• In a Hilbert space (infinite-dimensional), an orthonormal basis (Hilbert basis) requires the linear span to be dense; such bases exist by Zorn's lemma and allow Fourier-style expansions.
• Parseval's identity states that the squared norm of a vector equals the sum of squared Fourier coefficients with respect to an ONB.
• The set of all orthonormal bases for R^n is a principal homogeneous space (G-torsor) for the orthogonal group O(n), i.e., the Stiefel manifold V_n(R^n).
• A Hilbert space is separable if and only if it admits a countable orthonormal basis.

Connections: Orthonormal Basis · Inner Product Space · Hilbert Space · Gram Schmidt Process

Source: http://en.wikipedia.org/wiki/Orthonormal_basis