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About the Banach envelope of $$l_{1,\infty}$$. (English) Zbl 1175.46004
The author studies the Banach envelope $$l_{1,\infty}^{\text{ban}}$$ of the quasi-Banach space $$l_{1,\infty}$$ consisting of all sequences $$x = (\xi_k)$$ with $$s_n(x) = O(\frac 1 n)$$, where $$(s_n(x))$$ denotes the non-increasing rearrangement of $$x = (\xi _k)$$. The situation turns out to be much more complicated than that in the well-known case of the separable subspace $$l_{1,\infty}^\circ$$, whose members are characterized by $$s_n(x)=o(\frac 1 n)$$. Namely, the Banach envelope of the latter space is known to be $$m_{1,\infty}^\circ$$, the closed hull of all sequences with only a finite number of nonzero elements in the Sargent space $$m_{1,\infty}$$ which is the collection of all $$x=(\xi_k)$$ for which $$\| x | m_{1,\infty} \|:= \sup_n \frac{s_1(x)+ \ldots +s_n(x)}{\frac{1}{1} + \ldots +\frac{1}{n}}$$ is finite. However, as the main result the author proves that the norms $$\| \cdot | m_{1,\infty} \|$$ and $$\| \cdot | {l_{1,\infty}^{\text{ban}}} \|$$ fail to be equivalent on $$l_{1,\infty}$$. The proof uses an explicit formula for the norm $$\| \cdot | {l_{1,\infty}^{\text{ban}}} \|$$ induced on $$l_{1,\infty}$$ given in [N. J. Kalton and F. A. Sukochev, J. Reine Angew. Math. 621, 81–121 (2008; Zbl 1152.47014)]. For the convenience of the reader, the author provides an elementary proof of the inequality needed for the proof of the main result.

##### MSC:
 46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46B45 Banach sequence spaces
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