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Riesz sequence

In mathematics, a sequence of vectors (x n ) in a Hilbert space (H,\langle\cdot,\cdot\rangle) is called a Riesz sequence if there exist constants 0 such that 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 \{a_n\} and hence, for all \{a_n\}_{n=1}^{\infty}\in \ell^{2} . A Riesz sequence is called a Riesz basis if \overline{\mathop{\rm span} (x_n)} = H. Equivalently, a Riesz basis for H is a family of the form \left\{x_{n} \right\}_{n=1}^{\infty} = \left\{ Ue_{n} \right\}_{n=1}^{\infty} , where \left\{e_{n} \right\}_{n=1}^{\infty} is an orthonormal basis for H and U : H \rightarrow H is a bounded bijective operator. Subsequently, there exist constants 0 such that 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.

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Article summary

In mathematics, a sequence of vectors (x n ) in a Hilbert space (H,\langle\cdot,\cdot\rangle) is called a Riesz sequence if there exist constants 0 such that 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 \{a_n\} and hence, for all \{a_n\}_{n=1}^{\infty}\in \ell^{2} . A Riesz sequence is called a Riesz basis if \overline{\mathop{\rm span} (x_n)} = H. Equivalently, a Riesz basis for H is a family of the form \left\{x_{n} \right\}_{n=1}^{\infty} = \left\{ Ue_{n} \right\}_{n=1}^{\infty} , where \left\{e_{n} \right\}_{n=1}^{\infty} is an orthonormal basis for H and U : H \rightarrow H is a bounded bijective operator. Subsequently, there exist constants 0 such that 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.

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