1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ I should clarify that it only applies to decreasing sequences. $\endgroup$ – Tom Solberg Jan 20 at 7:59
  • $\begingroup$ Are you allowing the $a_i$ to be negative? $\endgroup$ – Yemon Choi Jan 20 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. $\endgroup$ – Bill Johnson Jan 20 at 17:57
  • $\begingroup$ Thanks @BillJohnson, I'll look into that. Yemon, I edited the question to ask the question more accurately. $\endgroup$ – Tom Solberg Jan 20 at 18:23
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.