Dense-lineability in spaces of continuous functions
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- by L. Bernal-González PDF
- Proc. Amer. Math. Soc. 136 (2008), 3163-3169 Request permission
Abstract:
In this paper we provide a general method to prove that certain nonlinear families of continuous functions contain dense linear manifolds. An application is furnished.References
- Richard Aron, V. I. Gurariy, and J. B. Seoane, Lineability and spaceability of sets of functions on $\Bbb R$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803. MR 2113929, DOI 10.1090/S0002-9939-04-07533-1
- Richard M. Aron, David Pérez-García, and Juan B. Seoane-Sepúlveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (2006), no. 1, 83–90. MR 2261701, DOI 10.4064/sm175-1-5
- Frédéric Bayart, Topological and algebraic genericity of divergence and universality, Studia Math. 167 (2005), no. 2, 161–181. MR 2134382, DOI 10.4064/sm167-2-4
- L. Bernal-González, Lineability of sets of nowhere analytic functions, J. Math. Anal. Appl. 340 (2008), 1284–1295.
- R. B. Darst, Most infinitely differentiable functions are nowhere analytic, Canad. Math. Bull. 16 (1973), 597–598. MR 346112, DOI 10.4153/CMB-1973-098-3
- Paul du Bois-Reymond, Ueber den Gültigkeitsbereich der Taylor’schen Reihenentwickelung, Math. Ann. 21 (1883), no. 1, 109–117 (German). MR 1510189, DOI 10.1007/BF01442615
- P. Enflo and V.I. Gurariy, On lineability and spaceability of sets in function spaces, preprint.
- V. P. Fonf, V. I. Gurariy, and M. I. Kadets, An infinite dimensional subspace of $C[0,1]$ consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), no. 11-12, 13–16. MR 1738120
- F. J. García-Pacheco, N. Palmberg, and J. B. Seoane-Sepúlveda, Lineability and algebrability of pathological phenomena in analysis, J. Math. Anal. Appl. 326 (2007), no. 2, 929–939. MR 2280953, DOI 10.1016/j.jmaa.2006.03.025
- V. I. Gurariĭ, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR 167 (1966), 971–973 (Russian). MR 0199674
- V. I. Gurariĭ, Linear spaces composed of everywhere nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13–16 (Russian). MR 1127022
- Vladimir I. Gurariy and Lucas Quarta, On lineability of sets of continuous functions, J. Math. Anal. Appl. 294 (2004), no. 1, 62–72. MR 2059788, DOI 10.1016/j.jmaa.2004.01.036
- Stanislav Hencl, Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3505–3511. MR 1707147, DOI 10.1090/S0002-9939-00-05595-7
- John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028
- M. Lerch, Ueber die Nichtdifferentiirbarkeit bewisser Funktionen, J. Reine Angew. Math. 103 (1888), 126–138.
- Dietrich Morgenstern, Unendlich oft differenzierbare nicht-analytische Funktionen, Math. Nachr. 12 (1954), 74 (German). MR 64830, DOI 10.1002/mana.19540120106
- John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443, DOI 10.1007/978-1-4684-9339-9
- L. Rodríguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3649–3654. MR 1328375, DOI 10.1090/S0002-9939-1995-1328375-8
- Helmut Salzmann and Karl Zeller, Singularitäten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354–367 (German). MR 71479, DOI 10.1007/BF01180644
- A.C.M. Van Rooij and W.H. Schikhoff, “A Second Course on Real Functions”, Cambridge University Press, Cambridge, 1982.
Additional Information
- L. Bernal-González
- Affiliation: Facultad de Matemáticas, Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, Avda. Reina Mercedes, Sevilla-41080, Spain
- Email: lbernal@us.es
- Received by editor(s): June 19, 2007
- Published electronically: April 25, 2008
- Additional Notes: The author has been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127, and by MEC Grants MTM2006-13997-C02-01 and Acción Especial MTM2004-21420-E
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3163-3169
- MSC (2000): Primary 46E10; Secondary 26A16, 26A27, 26E10
- DOI: https://doi.org/10.1090/S0002-9939-08-09495-1
- MathSciNet review: 2407080