I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is hard to do, but such $x$ exists by an indirect method (using Birkhorff transitivity theorem), see Dynamics of Linear Operators by Bayart and Matheron, p. 6.

  • $\begingroup$ The gliding hump. A method used in Banach sequence spaces a lot. $\endgroup$ – GEdgar Jul 16 '13 at 12:35
  • $\begingroup$ What about ordering the countable dense set of rational and eventually null sequences properly weighted into a sequence? $\endgroup$ – Jochen Jul 16 '13 at 13:04
  • 2
    $\begingroup$ Here is the article of S. Rolewicz where this was first proved. An elementary and constructive proof is contained therein. $\endgroup$ – David Mitra Jul 16 '13 at 23:10

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