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](https://mathoverflow.net/questions/288966/the-euclidean-norm-and-k-largest-elements), 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?



Characterizing a norm on sequences
The spaces you are asking about are called `Marcinkiewicz sequence spaces', see, for example <http://mate.dm.uba.ar/~slassall/marcinkiewicz.pdf>


