In mathematics, a sequence of vectors (xn) in a Hilbert space is called a Riesz sequence if there exist constants such that for all sequences of scalars (an) in the ℓp space ℓ2. A Riesz sequence is called a Riesz basis if . Theorems If H is a finite-dimensional space, then every basis of H is a Riesz basis. Let be in the Lp space L2(R), let and let denote the Fourier transform of . Define constants c and C with . Then the following are equivalent: The first of the above conditions is the definition for () to form a Riesz basis for the space it spans. See also Orthonormal basis Hilbert space Frame of a vector space References Functional analysis