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Riesz sequence
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(Definition)
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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.
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"Riesz sequence" is owned by swiftset.
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Cross-references: basis, finite-dimensional, scalars, Hilbert space, vectors, sequence
There is 1 reference to this entry.
This is version 4 of Riesz sequence, born on 2004-06-24, modified 2005-08-11.
Object id is 5963, canonical name is RieszSequence.
Accessed 5734 times total.
Classification:
| AMS MSC: | 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology ) |
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Pending Errata and Addenda
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