// Workers AI · dad joke modeWhat did the cyclic vector say? "I'm always turning back.

In the mathematics of operator theory, an operator $A$ on an (infinite-dimensional) Banach space or Hilbert space $H$ has a cyclic vector $f$ if the vectors $f,Af,A^{2}f,\dots$ span $H$. 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 polynomials, is dense in $H$.

See also

• Cyclic and separating vector

References

• Halmos, Paul R. (1982). "Cyclic vectors". A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19. New York: Springer. pp. 86–89. doi:10.1007/978-1-4684-9330-6_18. ISBN 978-1-4684-9332-0.