A General Topology Workbook by Iain T. Adamson
By Iain T. Adamson
This paintings goals to provide basic topology in an unconventional means. It supplies a evaluate of the elemental definitions including workouts with no strategies or proofs of the theorems partly 1, after which offers the ideas partly 2, permitting the scholar to match solutions with their very own.
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For references and further information, see [Tana14]. Surprisingly, the next two-dimensional problem, a particular case of Problem 86, remains apparently open. Problem 87. X/ D 2? We mention [Tana14], where a history of these problems and a list of references are included. We finish this section by mentioning in passing that K. Jarosz showed that every real Banach space X can be equivalently renormed so that all linear isometries of X onto itself are ˙ IdX [Ja88]. , [HMVZ08, p. 297]. x/ x0 k < ".
Study further the property of being approximable. , [FHHMZ11, Chap. 16]. 9 Extension of Operators Let X be a Banach space, let E be a subspace of X, and let 1. K/-EP, for short) if for every compact space K, any bounded linear operator T W E ! K/ admits an O Ä kTk. E; X/ has the extension TO W X ! K/-EP for some 1. Problem 66 (N. J. Kalton). K/-EP? We refer to [Kal06, p. 167]. It was proved by W. B. Johnson and M. K/-EP for every " > 0, but not for " D 0. We refer to [Zip03]. Problem 67. K/-EP?
Assume that X is a Banach space having the SCP. Let @ be an arbitrary cardinal number. Is it true that for an arbitrary subspace W of X with density @ there is a subspace V of X with V W and such that V is complemented in X and has density @? We refer to [PliYo00]. Related to Problem 36, we note that the following is open: 18 1 Basic Linear Structure Problem 37 (A. Plichko, D. Yost). Let X be a nonseparable Banach space with the Radon–Nikodým property. Does there exist a projection P on X with separable range?