By Vladimir V. Tkachuk

This fourth quantity in Vladimir Tkachuk's sequence on *Cp*-theory supplies kind of entire assurance of the speculation of practical equivalencies via 500 conscientiously chosen difficulties and routines. via systematically introducing all of the significant subject matters of *Cp*-theory, the ebook is meant to deliver a devoted reader from uncomplicated topological ideas to the frontiers of recent examine. The publication provides whole and up to date details at the renovation of topological houses via homeomorphisms of functionality areas. An exhaustive idea of *t*-equivalent, *u*-equivalent and *l*-equivalent areas is constructed from scratch. The reader also will locate introductions to the speculation of uniform areas, the idea of in the neighborhood convex areas, in addition to the speculation of inverse structures and measurement concept. furthermore, the inclusion of Kolmogorov's resolution of Hilbert's challenge thirteen is integrated because it is required for the presentation of the idea of *l*-equivalent areas. This quantity comprises crucial classical effects on useful equivalencies, specifically, Gul'ko and Khmyleva's instance of non-preservation of compactness via *t*-equivalence, Okunev's approach to developing *l*-equivalent areas and the theory of Marciszewski and Pelant on *u*-invariance of absolute Borel sets.

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**Example text**

L; / is a linear topological space and B is a local base of at 0. 203. Let L be a linear topological space. B/ is also balanced; (6) if E is an l-bounded subset of L then E is also l-bounded. 204. Let L be a linear topological space. Prove that (1) every neighborhood of 0 contains an open balanced neighborhood of 0; (2) every convex neighborhood of 0 contains an open convex balanced neighborhood of 0. Deduce from (2) that any locally convex linear topological space has a local base B at 0 such that each U 2 B is convex and balanced.

L=N is linear, open, and continuous; (3) If P 2 fmetrizability, local convexity, normability, complete normabilityg and L has P then L=N also has P. 213. Prove that any product of locally convex spaces is a locally convex space. 214. Suppose that L and M are linear topological spaces and ˚ is an equicontinuous family of linear maps from L to M . A/ B for all f 2 ˚. 215. Suppose that L and M are linear topological spaces and ˚ is a family of linear continuous maps from L to M . x/ is l-bounded in M g is of second category in L.

Suppose that L and M are linear topological spaces and ˚ is a family of linear continuous maps from L to M . x/ is l-bounded in M g is of second category in L. Prove that B D L and the family ˚ is equicontinuous. 216. (Hahn–Banach theorem) Let L be a linear space (without topology). Suppose that we are given a map p W L ! x/ for all x; y 2 L and t 0. Prove that, for any linear subspace M of the linear space L and any linear functional f W M ! x/ for any x 2 M , there exists a linear functional F W L !