# Characterizing a norm on sequences

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?

• I should clarify that it only applies to decreasing sequences. Jan 20, 2019 at 7:59
• Are you allowing the $a_i$ to be negative? Jan 20, 2019 at 13:51
• 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. Jan 20, 2019 at 17:57
• Thanks @BillJohnson, I'll look into that. Yemon, I edited the question to ask the question more accurately. Jan 20, 2019 at 18:23