Riesz sequence

Definition

A sequence of vectors $(x_{n})$ in a Hilbert space $H$ is called a Riesz sequence if there exist constants $0<c\leq C$ such that

c\left(\sum_{n}|a_{n}|^{2}\right)\leq\left\|\sum_{n}a_{n}x_{n}\right\|^{2}\leq C% \left(\sum_{n}|a_{n}|^{2}\right)

for all sequences of scalars $(a_{n})\in l^{2}$ . A Riesz sequence is called a Riesz basis if $\overline{\mathop{\rm span}(x_{n})}=H$ .

If $H$ is a finite-dimensional space, then every basis of $H$ is a Riesz basis.

Title	Riesz sequence
Canonical name	RieszSequence
Date of creation	2013-03-22 14:26:53
Last modified on	2013-03-22 14:26:53
Owner	swiftset (1337)
Last modified by	swiftset (1337)
Numerical id	7
Author	swiftset (1337)
Entry type	Definition
Classification	msc 46C05
Related topic	OrthonormalBasis
Related topic	HilbertSpace
Related topic	Frame2
Related topic	AnEquivalentConditionForTheTranslatesOfAnL_2FunctionToFormARieszBasis
Defines	Riesz basis