I'm trying to solve an exercise about Lorentz sequence spaces. Below is the text.

It is known that the Lorentz sequence space, denoted by $\ell\left(p,q\right) $ is the set of all sequences $ a = \left\lbrace a_n\right\rbrace $ such that the functional $ \Vert a\Vert_{pq} < \infty, $ where

$\Vert a\Vert_{pq} =\begin{cases} \left(\sum_{n=1}^{\infty} \left(n^{\frac{1}{p}} a_{n}^{\ast}\right)^{q} n^{-1}\right)^{\frac{1}{q}}\\ \sup_{n\geq 1} n^{\frac{1}{p}} a^{\ast}_{n} \end{cases}$

if $ 0<p\leq\infty, 0<q<\infty $ and $ 0<p\leq\infty, q = \infty $ respectively. Moreover $ a^{\ast} = \left\lbrace a^{\ast}_{n}\right\rbrace $ is the sequence $ \left\lbrace\vert a_{n}\vert\right\rbrace $ permutated in a decreasing order.

Now consider the two-dimensional Lorentz sequence space $ \ell^{\left( 2\right)}\left(p,q\right), $ that is the space of all sequences $ a = \left\lbrace a_{1}, a_{2}\right\rbrace $ with the quasi-norm $$ \Vert a\Vert_{pq} = \left( a_{1}^{\ast^{q}} + 2^{\frac{q}{p} -1} a_{2}^{\ast^{q}}\right)^{\frac{1}{q}}. $$

Prove that $ \Vert\cdot\Vert_{pq} $ cannot be a norm for $ \ell^{\left( 2\right)}\left(p, q\right) $ when $ p < q. $

Hint: Consider the unit ball in $ \ell^{\left(2\right)}\left(p, q\right) $ for different values of $ p $ and $ q. $

It's clear that it can be shown as done for the standard Lorentz spaces, but (about me) the author chooses $\ell^{\left( 2\right)}\left(p,q\right)$ because, since the dimension is $2,$ the unit ball can be drawn in the plane. My idea is to prove that, in the case $ p < q, $ the unit ball is not convex so $ \Vert\cdot\Vert_{pq} $ fails to be a norm. Is that right what i said until now?

For $ p = q = 1, $ we have $$ \Vert a\Vert_{11} = \vert a_1\vert + \vert a_2\vert \leq 1, $$ that is a rhombus, that is convex and then $ \Vert\cdot\Vert_{11} $ is a norm, as expected. I would like to prove that, for example, for $ p = 1, q = \infty, $ the $ \Vert\cdot\Vert_{1\infty} $ is not a norm, but i don't know how to express the norm in this case, i.e. what is the value of $ \Vert a\Vert_{1\infty}? $

I expect to draw a not covex figure in the plane, but i don't know how.

I hope everyone could help me! Thank you!

  • 1
    $\begingroup$ I can at least verify the result is correct graphically on Desmos desmos.com/calculator/f8vn8we6h7. Coming to the result caveman style is another thing... $\endgroup$ – Calvin Khor Oct 12 '19 at 8:25
  • $\begingroup$ PS I'd expect in analogy with the full sequence norm, $$\|a\|_{1\infty} = \max\{ a^*_1, 2^{1/p}a^*_2\} $$ for this space, $(1/2,1)$ and $(1/2,1)$ are in the (closed) unit ball, but their average $(3/4,3/4)$ is not. $\endgroup$ – Calvin Khor Oct 12 '19 at 8:36
  • $\begingroup$ I'm sorry but i thought I'd solve it like a caveman. You offered a very good way to solution. $\endgroup$ – C. Bishop Oct 13 '19 at 8:11
  • $\begingroup$ I think the method will work eventually, but I don't consider the problem solved. I would like to see the full "caveman" answer :) or maybe someone has a slick method that we are missing...so I don't agree with accepting my answer. $\endgroup$ – Calvin Khor Oct 13 '19 at 8:30

(the below gets the result for $p\ll q$. I don't have time for the full result, but hope this is helpful anyway...)

In analogy with the full sequence norm, I'd say that $$|a|_{p\infty} := \max\{ a^*_1, 2^{1/p}a^*_2\}. $$ For this space, $(1/2^{1/p},1)$ and $(1/2^{1/p},1)$ are in the (closed) unit ball, but their average is not. Indeed, the average has norm $$ \frac{1/2^{1/p}+1}2 |(1,1)|_{p\infty} = \frac{2^{1/p}(1/2^{1/p}+1)}2 = \frac{2^{1/p} + 1}2 > 1.$$

Hoping that this works naively for other $q$. Let $p<q$. This translates to $2^{q/p-1}>1$. Set $a\le 1$, and consider $$x_1 = (1,a), \quad x_2 = (a,1).$$ Their norms are $$|x_1|_{pq} =|x_2|_{pq} = (1+a^q2^{q/p-1})^{1/q}$$ Setting $x_3 = (x_1 + x_2)/2 = (\frac{1+a}2,\frac{1+a}2) = \frac{1+a}2(1,1)$, $$ |x_3|_{pq} = \frac{1+a}2 |(1,1)|_{pq} = \frac{1+a}2(1+2^{q/p-1})^{1/q}$$ Convexity of the (closed) ball of radius $|x_1|_{pq}$ (or equivalently any ball of radius 1) would of course imply that $$ |x_3|_{pq} \le \frac12 |x_1|_{pq} + \frac12 |x_2|_{pq} = (1+a^q2^{q/p-1})^{1/q}$$

So the question is - is it true that for $a\in[0,1]$, $$ \frac{1+a}2(1+2^{q/p-1})^{1/q} \overset{\Huge ?}{\le} (1+a^q2^{q/p-1})^{1/q}$$ Setting $a = 2^{1/q - 1/p}<1$ in analogy with the $p\infty$ case. This inequality is

$$ \frac{1+2^{1/q - 1/p}}2(1+2^{q(1/p-1/q)})^{1/q} \overset{\Huge ?}{\le} 2^{1/q}$$

As noted, this inequality is false for $q\to\infty$. Its true at $p=q$. It looks likely to work on the graph(see https://www.desmos.com/calculator/qg2cpfocrj), but I don't have the time now to persue this further.


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