Diametral dimensions of Fréchet spaces. (English) Zbl 1361.46003
B. S. Mityagin [Russ. Math. Surv. 16, No. 4, 59–128 (1961); translation from Usp. Mat. Nauk 16, No. 4(100), 63–132 (1961; Zbl 0104.08601)] utilized Kolmogorov diameters to introduce a topological invariant of locally convex spaces \(X\) called the diametral dimension \(\Delta(X)\). He also implicitly studied another invariant \(\Delta_b(X)\), which always contains \(\Delta(X)\). These two diametral dimensions coincide for each non-Montel Fréchet space and they coincide with the Banach space \(c_0\) of null sequences. A. T. Terz{i}oğlu investigated in [Turk. J. Math. 37, No. 5, 847–851 (2013; Zbl 1287.46003)] the equality \(\Delta(X)=\Delta_b(X)\) for Fréchet-Schwartz spaces.
In the article under review, the authors introduce two new, larger variants of the diametral dimensions, replacing null sequences by bounded sequences in the definitions, and they show that if \(X\) is a Fréchet-Schwartz space, then \(\Delta^{\infty}(X)=\Delta_b^{\infty}(X)\). They then prove that the equality \(\Delta(X)=\Delta_b(X)\) holds for hilbertizable Fréchet-Schwartz spaces, in particular for nuclear Fréchet spaces. The proof is based on a result about the Kolmogorov diameters of the composition of two compact operators on Hilbert spaces.
In the last section, the authors extend several results of {Terz{i}oğlu} about prominent bounded sets. They prove that if a Fréchet space \(X\) has the topological invariant \((\overline{\Omega})\) of Vogt and Wagner, then \(X\) has a prominent bounded set. The converse result holds for regular Köthe echelon spaces. However, the equivalence does not hold in general. In fact, \(X=H(\mathbb{D}) \times H(\mathbb{C})\) has a prominent bounded set, but it does not satisfy \((\overline{\Omega})\).
In the article under review, the authors introduce two new, larger variants of the diametral dimensions, replacing null sequences by bounded sequences in the definitions, and they show that if \(X\) is a Fréchet-Schwartz space, then \(\Delta^{\infty}(X)=\Delta_b^{\infty}(X)\). They then prove that the equality \(\Delta(X)=\Delta_b(X)\) holds for hilbertizable Fréchet-Schwartz spaces, in particular for nuclear Fréchet spaces. The proof is based on a result about the Kolmogorov diameters of the composition of two compact operators on Hilbert spaces.
In the last section, the authors extend several results of {Terz{i}oğlu} about prominent bounded sets. They prove that if a Fréchet space \(X\) has the topological invariant \((\overline{\Omega})\) of Vogt and Wagner, then \(X\) has a prominent bounded set. The converse result holds for regular Köthe echelon spaces. However, the equivalence does not hold in general. In fact, \(X=H(\mathbb{D}) \times H(\mathbb{C})\) has a prominent bounded set, but it does not satisfy \((\overline{\Omega})\).
Reviewer: José Bonet (Valencia)
MSC:
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
| 46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |
| 46A63 | Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces |
References:
| [1] | [BD16]F. Bastin and L. Demeulenaere, On the equality between two diametral dimensions, Funct. Approx. Comment. Math., to appear (2016). · Zbl 1409.46006 |
| [2] | [MV97]R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Grad. Texts Math. 2, Oxford Univ. Press, New York, 1997. 280L. Demeulenaere et al. [Mit61]B. S. Mityagin, Approximate dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (1961), no. 4 (100), 63–132 (in Russian), English transl.: Russian Math. Surveys 16 (1961), no. 4, 59–127. [Pe l57]A. Pe lczy’nski, On the approximation of S-spaces by finite dimensional spaces, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 879–881. |
| [3] | [PCB87]P. P’erez Carreras and J. Bonet, Barrelled Locally Convex Spaces, NorthHolland Math. Stud. 131, North-Holland, Amsterdam, 1987. [Pin85]A. Pinkus, n-Widths in Approximation Theory, Ergeb. Math. Grenzgeb. 7, Springer, Berlin, 1985. [Ter08]T. Terzioglu, Diametral dimension and K”othe spaces, Turkish J. Math. 32 (2008), 213–218. [Ter13]T. Terzioglu, Quasinormability and diametral dimension, Turkish J. Math. 37 (2013), 847–851. [Vog00]D. Vogt, Lectures on Fr’echet spaces, Tech. report, Bergische Universit”at Wuppertal, 2000, www2.math.uni-wuppertal.de/ogt/vorlesungen/fs.pdf. [Wag80]M.-J. Wagner, Quotientenr”aume von stabilen Potenzreihenr”aumen endlichen Typs, Manuscripta Math. 31 (1980), 97–109. [Zah73]V. P. Zahariuta, On the isomorphism of cartesian products of locally convex spaces, Studia Math. 46 (1973), 201–221. Lo”ıc Demeulenaere |
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