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Riesz Sequence

is called a Riesz sequence if there exist constants $0 < c \leq C$ such that

$\displaystyle c\left( \sum_n \vert a_n\vert^2 \right) \leq \left\Vert \sum_n a_n x_n \right\Vert^2 \leq C \left( \sum_n \vert a_n\vert^2 \right)$

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