Riesz sequence

In mathematics, a sequence of vectors (x_n) in a Hilbert space ${\displaystyle (H,\langle \cdot ,\cdot \rangle )}$ is called a Riesz sequence if there exist constants ${\displaystyle 0<c\leq C<\infty }$ such that $${\displaystyle c\sum _{n=1}^{\infty }|a_{n}|^{2}\leq \left\Vert \sum _{n=1}^{\infty }a_{n}x_{n}\right\Vert ^{2}\leq C\sum _{n=1}^{\infty }|a_{n}|^{2},}$$ for every finite scalar sequence ${\displaystyle \{a_{n}\}}$ and hence, for all ${\displaystyle \{a_{n}\}_{n=1}^{\infty }\in \ell ^{2}}$.

A Riesz sequence is called a Riesz basis if $${\displaystyle {\overline {\mathop {\rm {span}} (x_{n})}}=H.}$$ Equivalently, a Riesz basis for ${\displaystyle H}$ is a family of the form ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty }=\left\{Ue_{n}\right\}_{n=1}^{\infty }}$, where ${\displaystyle \left\{e_{n}\right\}_{n=1}^{\infty }}$ is an orthonormal basis for ${\displaystyle H}$ and ${\displaystyle U:H\rightarrow H}$ is a bounded bijective operator. Subsequently, there exist constants ${\displaystyle 0<c\leq C<\infty }$ such that $${\displaystyle c\|f\|^{2}\leq \sum _{n=1}^{\infty }|\langle f,x_{n}\rangle |^{2}\leq C\|f\|^{2},\quad \forall f\in H.}$$ Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.