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}\\leqC%\\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.