A quantitative version of Krein’s theorems for Fréchet spaces
详细信息 查看全文
- 作者：Carlos Angosto (1)
Jerzy Ka?kol (2)
Albert Kubzdela (3)
Manuel López-Pellicer (4) - 关键词：Primary 46A50 ; Secondary 54C35 ; Krein’s theorem ; Compactness ; Fréchet space ; Space of continuous functions
- 刊名：Archiv der Mathematik
- 出版年：2013
- 出版时间：July 2013
- 年：2013
- 卷：101
- 期：1
- 页码：65-77
- 全文大小：253KB
- 参考文献：1. Angosto C., Cascales B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156, 1412-421 (2009) CrossRef
2. C. Angosto, Distance to spaces of functions, PhD thesis, Universidad de Murcia (2007).
3. C. Angosto and B. Cascales, A new look at compactness via distances to functions spaces, World Sc. Pub. Co. (2008).
4. Angosto C., Cascales B.: The quantitative difference between countable compactness and compactness. J. Math. Anal. Appl. 343, 479-91 (2008) CrossRef
5. Angosto C., Cascales B., Namioka I.: Distances to spaces of Baire one functions. Math. Z. 263, 103-24 (2009) CrossRef
6. C. Angosto, J. Ka?kol, and M. López-Pellicer, A quantitative approach to weak compactness in Fréchet spaces and spaces / C( / X), J. Math. Anal. Appl. 403 (2013), 13-2.
7. Cascales B., Marciszesky W., Raja M.: Distance to spaces of continuous functions. Topology Appl. 153, 2303-319 (2006) CrossRef
8. M. Fabian et?al. Functional Analysis and Infinite-dimensional geometry, CMS Books in Mathematics, Canadian Math. Soc., Springer (2001).
9. M. Fabian et?al. A quantitative version of Krein’s theorem, Rev. Mat. Iberoam. 21 (2005), 237-48
10. Granero A. S.: An extension of the Krein-Smulian Theorem. Rev. Mat. Iberoam. 22, 93-10 (2006) CrossRef
11. Granero A. S., Hájek P., Montesinos V.: Santalucía, Convexity and / ω*-compactness in Banach spaces. Math. Ann. 328, 625-31 (2004) CrossRef
12. Grothendieck A.: Criteres de compacité dans les spaces fonctionnelles généraux. Amer. J. Math. 74, 168-86 (1952) CrossRef
13. Khurana S. S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2, 721-24 (1979) CrossRef - 作者单位：Carlos Angosto (1)
Jerzy Ka?kol (2)
Albert Kubzdela (3)
Manuel López-Pellicer (4)
1. Depto. de Matemática Aplicada y Estadistica, Universidad Politécnica de Cartagena, 30203, Cartagena, Spain
2. Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614, Poznan, Poland
3. Institute of Civil Engineering, Poznań University of Technology, Ul. Piotrowo 5, 61-138, Poznan, Poland
4. Depto. de Matemática Aplicada and IUMPA, Universitat Politècnica de València, 46022, Valencia, Spain
文摘
For a Banach space E and its bidual space E ′-/sup>, the following function ${k(H) : = {\rm sup}_{y\in\overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}} {\rm inf}_{x\in E} \|y - x\|}$ defined on bounded subsets H of E measures how far H is from being σ(E, E--relatively compact in E. This concept, introduced independently by Granero [10] and Cascales et?al. [7], has been used to study a quantitative version of Krein’s theorem for Banach spaces E and spaces C p (K) over compact K. In the present paper, a quantitative version of Krein’s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E the above function k(H) reads as follows ${k(H) := {\rm sup}\{d(h, E) : h \in \overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}\},}$ where d(h, E) is the natural distance of h to E in the bidual E ′-/sup>. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds ${k(coH) < (2^{n+1} - 2) k(H) + \frac{1}{2^{n}}}$ for all ${n \in \mathbb{N}}$ . Consequently this yields also the following formula ${k(coH) \leq \sqrt{k(H)}(3 - 2\sqrt{k(H)})}$ . Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein’s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet space. We also define and discuss two other measures of weak non-compactness lk(H) and k-H) for a Fréchet space and provide two quantitative versions of Krein’s theorem for both functions.