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?

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

decreasingsequences. $\endgroup$ – Tom Solberg Jan 20 at 7:59