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Hilbert-generated spaces. (English) Zbl 1039.46015
A Banach space $$X$$ is called $$\mathcal P$$-generated (where $$\mathcal P$$ is a property of Banach spaces) if there is a Banach space $$Y$$ with property $$\mathcal P$$ and a continuous linear operator from $$Y$$ into $$X$$ with dense range. M. Fabian, G. Godefroy and V. Zizler [Isr. J. Math. 124, 243–252 (2001; Zbl 1027.46012)] showed that a Banach space is isomorphic to a subspace of a Hilbert-generated space if and only if it admits an equivalent uniformly Gâteaux smooth norm. The present paper contains a more detailed study of the class of subspaces of Hilbert-generated spaces.
There is a natural linear hierarchy of subclasses of this class. Six classes are mentioned, the smallest one is that of Hilbert-generated spaces followed by that of super-reflexive-generated ones and the largest one is that of subspaces of Hilbert-generated spaces. All the inclusions between these classes are shown to be proper using counterexamples (note that one of these examples uses the continuum hypothesis).
Further results are devoted to characterizations of some of theses subclasses within spaces of density $$\aleph_1$$. In particular, $$\ell_p(\Gamma)$$-generated spaces for $$1< p\leq2$$ are characterized by renorming with a norm satisfying a quantitative version of uniform Gâteaux smoothness. It is an open question whether these characterizations are valid also for spaces with larger densities.

MSC:
 46B26 Nonseparable Banach spaces 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory
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References:
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