On multi-orthogonal bases in finite-dimensional non-Archimedean normed spaces

AbstractThe paper deals with the problem of the existence multi-orthogonal bases in finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field

A quantitative version of Krein's theorems for Fréchet spaces

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being &#963;(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(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) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. 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) &#8722; 2)k(H) + 1/2^n for all n &#8712; N. Consequently, this yields also the following formula k(coH) &#8804; (k(H))^(1/2))(3-2(k(H)^(1/2))). 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 spaces. 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.The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.Angosto Hernández, C.; Kakol, J.; Kubzdela, A.; López Pellicer, M. (2013). A quantitative version of Krein's theorems for Fréchet spaces. Archiv der Mathematik. 101(1):65-77. https://doi.org/10.1007/s00013-013-0513-4S65771011Angosto C., Cascales B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156, 1412–1421 (2009)C. Angosto, Distance to spaces of functions, PhD thesis, Universidad de Murcia (2007).C. Angosto and B. Cascales, A new look at compactness via distances to functions spaces, World Sc. Pub. Co. (2008).Angosto C., Cascales B.: The quantitative difference between countable compactness and compactness. J. Math. Anal. Appl. 343, 479–491 (2008)Angosto C., Cascales B., Namioka I.: Distances to spaces of Baire one functions. Math. Z. 263, 103–124 (2009)C. Angosto, J. Ka̧kol, and M. López-Pellicer, A quantitative approach to weak compactness in Fréchet spaces and spaces C(X), J. Math. Anal. Appl. 403 (2013), 13–22.Cascales B., Marciszesky W., Raja M.: Distance to spaces of continuous functions. Topology Appl. 153, 2303–2319 (2006)M. Fabian et al. Functional Analysis and Infinite-dimensional geometry, CMS Books in Mathematics, Canadian Math. Soc., Springer (2001).M. Fabian et al. A quantitative version of Krein’s theorem, Rev. Mat. Iberoam. 21 (2005), 237–248Granero A. S.: An extension of the Krein-Smulian Theorem. Rev. Mat. Iberoam. 22, 93–110 (2006)Granero A. S., Hájek P., Montesinos V.: Santalucía, Convexity and ω*-compactness in Banach spaces. Math. Ann. 328, 625–631 (2004)Grothendieck A.: Criteres de compacité dans les spaces fonctionnelles généraux. Amer. J. Math. 74, 168–186 (1952)Khurana S. S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2, 721–724 (1979

On non-Archimedean hilbertian spaces

AbstractWe consider a special class of non-Archimedean Banach spaces, called Hilbertian, for which every one-dimensional linear subspace has an orthogonal complement. We prove that all immediate extensions of co, contained in l∞, are Hilbertian. In this way we construct examples of Hilbertian spaces over a non-spherically complete valued field without an orthogonal base

A non-archimedean Dugundji extension theorem

summary:We prove a non-archimedean Dugundji extension theorem for the spaces $C^{\ast }(X,\mathbb {K})$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $\mathbb {K}$. Assuming that $\mathbb {K}$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T\colon C^{\ast }(Y,\mathbb {K})\rightarrow C^{\ast }(X,\mathbb {K})$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $\mathbb {K}$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular space $X$ is a retract of $X$