# Cyclic vector

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In the [mathematics](/source/mathematics) of [operator theory](/source/operator_theory), an [operator](/source/linear_operator) ''A'' on an ([infinite-dimensional](/source/dimension_(vector_space))) [Banach space](/source/Banach_space) or [Hilbert space](/source/Hilbert_space) '''H''' has a cyclic vector ''f'' if the vectors ''f'', ''Af'', ''A''<sup>2</sup>''f'',... [span](/source/spanning_set) '''H'''. Equivalently, ''f'' is a cyclic vector for ''A'' in case the set of all vectors of the form ''p''(''A'')''f'', where ''p'' varies over all [polynomial](/source/polynomial)s, is [dense](/source/dense_set) in '''H'''.<ref>{{cite book |title=A Hilbert Space Problem Book |volume= 19 |pages= 86–89 |doi= 10.1007/978-1-4684-9330-6_18 |chapter= Cyclic Vectors |series= Graduate Texts in Mathematics |year= 1982 |last1= Halmos |first1= Paul R. |isbn= 978-1-4684-9332-0 }}</ref><ref>{{cite web |title=Cyclic vector |url=http://www.encyclopediaofmath.org/index.php?title=Cyclic_vector&oldid=34882 |website=Encyclopedia of Mathematics}}</ref>

==See also==
* [Cyclic and separating vector](/source/Cyclic_and_separating_vector)

==References==
{{reflist}}

Category:Abstract algebra
Category:Functional analysis

{{Abstract-algebra-stub}}

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Adapted from the Wikipedia article [Cyclic vector](https://en.wikipedia.org/wiki/Cyclic_vector) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cyclic_vector?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.