Description of some monotone norms (Q2748960)

For every sequence \(x= (\xi_j)^\infty\), of real or complex numbers and \(n,m\in\mathbb{N}\), \(n< m\) let \(\sigma_{nm}(x)\) denote the sequence \((\xi_{n+1},\dots, \xi_m, 0,0,0,\dots)\). Let \(\widehat c\) be the linear space of such sequences of finite rank. A norm \(\|\cdot\|\) on \(\widehat c\) is said to be monotone if \(x= (\xi_j)^\infty_1\), \(y= (\eta_j)^\infty_1\) with \(|\xi_j|\leq |\eta_j|\) \((j\in\mathbb{N})\) implies that \(\|x\|\leq \|y\|\). The author proves that if \(\|\cdot\|\) is monotone, \(\|\ell_k\|= 1\) \((k\in\mathbb{N})\) and for every increasing sequence \((n_k)^\infty_0\) of natural numbers we have \(\|x\|= \|(\sigma_{n_{k-1}n_k}(x)\|)^\infty_1\|\), then there exists \(p\), \(1\leq p\leq\infty\), such that: NEWLINE\[NEWLINE\|x\|=\|x\|_p= \begin{cases} (\sum^\infty_{j=1} |\xi_j|^p)^{1/p}\quad &\text{if }1\leq p<\infty,\\ \sup_j|\xi_j|\quad &\text{if }p= \infty.\end{cases}.NEWLINE\]