By Jerzy Kąkol, Wieslaw Kubiś, Manuel López-Pellicer

ISBN-10: 1461405289

ISBN-13: 9781461405283

A huge mathematical neighborhood during the international actively works in useful research and makes use of profound strategies from topology. because the first monograph to process the subject of topological vector areas from the viewpoint of descriptive topology, this paintings presents additionally new insights into the connections among the topological homes of linear functionality areas and their function in practical research.

*Descriptive Topology in chosen themes of useful Analysis* is a self-contained quantity that applies contemporary advancements and classical ends up in descriptive topology to check the periods of infinite-dimensional topological vector areas that seem in useful research. Such areas comprise Fréchet areas, *LF*-spaces and their duals, and the gap of continuing real-valued features *C*(*X*) on a very commonplace Hausdorff house *X*, to call a couple of. those vector areas seem in distribution concept, differential equations, complicated research, and numerous different parts of sensible analysis.

Written by means of 3 specialists within the box, this ebook is a treasure trove for researchers and graduate scholars learning the interaction one of the components of point-set and descriptive topology, sleek research, set idea, topological vector areas and Banach areas, and non-stop functionality areas. in addition, it is going to function a reference for current and destiny paintings performed during this quarter and will function a precious complement to complex graduate classes in practical research, set-theoretic topology, or the speculation of functionality spaces.

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**Additional resources for Descriptive Topology in Selected Topics of Functional Analysis**

**Example text**

9 is applied to deduce that X is strongly realcompact. 9 and the previous case. 9, we know that a realcompact space X for which βX \ X is countably compact is strongly realcompact. 10, we note only the following fact. , its image under any real-valued continuous function is bounded). Proof Assume that X ∗ := βX \ X is not pseudocompact. Then there exists a locally finite sequence (Un )n of open disjoint subsets in X ∗ . Then, by regularity, we obtain a sequence (Vn )n of open nonempty sets in βX such that ∅ = Vn ∩ X ∗ ⊂ Vn ∩ X ∗ ⊂ Un , where the closure is taken in βX.

11 (Kakol–Saxon–Todd) ˛ Cc (X) does not contain a dense subspace RN if and only if X is Warner bounded. For the proof, we need the following lemma. 8 (a) An lcs E contains a dense subspace G of RN if and only if there exists a sequence (wn )n of nonzero elements in E such that every continuous seminorm in E vanishes at wn for almost all n ∈ N. (b) If an lcs E contains a dense subspace G of RN , then the strong dual (E , β(E , E)) contains the space ϕ. Proof (a) Assume that E contains a sequence as mentioned.

This yields that L ∪ A is closed. As every topologically bounded set in X is relatively compact, there exists a continuous function f ∈ C(X) unbounded on L ∪ A. Therefore f is unbounded on each An . (ii) Fix x ∈ X. By the assumption, there exists a compact set K, x ∈ K, and a decreasing basis (Un )n of open neighborhoods of K. Set Hn := f ∈ C(X) : sup |f (h)| ≤ n . h∈Un Then, as is easily seen, the family {Hn : n ∈ N} covers C(X). Since Cc (X) is Bairelike, there exist a compact set D ⊂ X, ε > 0, and n ∈ N such that f ∈ C(X) : sup |f (d)| < ε ⊂ Hn .

### Descriptive Topology in Selected Topics of Functional Analysis by Jerzy Kąkol, Wieslaw Kubiś, Manuel López-Pellicer

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