The p-envelope of an F-space is the p-convex analogue of the Fréchet envelope. We show that if an F-space is locally bounded (i.e., a quasi-Banach space) with separating dual, then the p-envelope coincides with the Banach envelope only if the space is already locally convex. By contrast, we give examples of F-spaces with are not locally bounded nor locally convex for which the p-envelope and the Fréchet envelope are the same.
@article{bwmeta1.element.bwnjournal-article-apmv57z2p121bwm,
author = {C. M. Eoff},
title = {p-Envelopes of non-locally convex F-spaces},
journal = {Annales Polonici Mathematici},
volume = {57},
year = {1992},
pages = {121-134},
zbl = {0777.46002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv57z2p121bwm}
}
C. M. Eoff. p-Envelopes of non-locally convex F-spaces. Annales Polonici Mathematici, Tome 57 (1992) pp. 121-134. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv57z2p121bwm/
[000] [1] A. B. Aleksandrov, Essays on non-locally convex Hardy classes, in: Complex Analysis and Spectral Theory, Lecture Notes in Math. 864, Springer, Berlin 1981, 1-89.
[001] [2] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in , Astérisque 77 (1980), 11-66. | Zbl 0472.46040
[002] [3] P. L. Duren, Theory of Spaces, Academic Press, New York 1970. | Zbl 0215.20203
[003] [4] P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on spaces with 0 < p < 1, J. Reine Angew. Math. 238 (1969), 32-60.
[004] [5] C. M. Eoff, Fréchet envelopes of certain algebras of analytic functions, Michigan Math. J. 35 (1988), 413-426.
[005] [6] N. J. Kalton, Analytic functions in non-locally convex spaces and applications, Studia Math. 83 (1986), 275-303. | Zbl 0634.46038
[006] [7] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space Sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, 1984.
[007] [8] J. E. McCarthy, Topologies on the Smirnov class, to appear.
[008] [9] N. Mochizuki, Algebras of holomorphic functions between and , Proc. Amer. Math. Soc. 105 (1989), 898-902.
[009] [10] M. Nawrocki, Linear functionals on the Smirnov class of the unit ball in , Ann. Acad. Sci. Fenn. AI 14 (1989), 369-379.
[010] [11] M. Nawrocki, The Fréchet envelopes of vector-valued Smirnov classes, Studia Math. 94 (1989), 163-177. | Zbl 0702.46021
[011] [12] J. W. Roberts and M. Stoll, Prime and principal ideals in the algebra N⁺, Arch. Math. (Basel) 27 (1976), 387-393; Correction, ibid. 30 (1978), 672. | Zbl 0329.46025
[012] [13] J. H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43 (1976), 187-202. | Zbl 0354.46036
[013] [14] J. H. Shapiro, Remarks on F-spaces of analytic functions, in: Banach Spaces of Analytic Functions, Lecture Notes in Math. 604, Springer, Berlin 1977, 107-124.
[014] [15] M. Stoll, Mean growth and Taylor coefficients of some topological algebras of analytic functions, Ann. Polon. Math. 35 (1977), 139-158. | Zbl 0377.30036
[015] [16] N. Yanagihara, Multipliers and linear functionals for the class N⁺, Trans. Amer. Math. Soc. 180 (1973), 449-461. | Zbl 0243.46036
[016] [17] N. Yanagihara, The containing Fréchet space for the class N⁺, Duke Math. J. 40 (1973), 93-103. | Zbl 0247.46040
[017] [18] A. I. Zayed, Topological vector spaces of analytic functions, Complex Variables 2 (1983), 27-50. | Zbl 0495.46019