Let $\{a_i\}$ be a sequence of reals such that $|a_i|\geq|a_{i+1}|$ for all $i$, and consider the following norm: $$\|\{a_i\}\| = \sup_k \frac{1}{\sqrt{k}}\sum_{i=1}^k |a_i|~.$$ One can see that -- among all decreasing sequences -- this norm is bounded above by the $\ell_2$ norm as explained here, and it is obviously bounded below by the $\ell_\infty$ norm (which just corresponds to the case $k=1$), although neither bound is uniformly tight. Are there any other norms that this is "similar" to?

  • $\begingroup$ I should clarify that it only applies to decreasing sequences. $\endgroup$ Jan 20, 2019 at 7:59
  • $\begingroup$ Are you allowing the $a_i$ to be negative? $\endgroup$
    – Yemon Choi
    Jan 20, 2019 at 13:51
  • 2
    $\begingroup$ Google "Lorentz sequence spaces". Since you are interested only in decreasing sequence, it is more natural to define the norm of a general sequence to be your norm of the decreasing rearrangement of the sequence. $\endgroup$ Jan 20, 2019 at 17:57
  • $\begingroup$ Thanks @BillJohnson, I'll look into that. Yemon, I edited the question to ask the question more accurately. $\endgroup$ Jan 20, 2019 at 18:23

1 Answer 1


The spaces you are asking about are called `Marcinkiewicz sequence spaces', see, for example http://mate.dm.uba.ar/~slassall/marcinkiewicz.pdf


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