On the dimension of subspaces of continuous functions attaining their maximum finitely many times

Bernal-González, L.; Cabana-Méndez, H.; Muñoz-Fernández, G.; Seoane-Sepúlveda, J.

doi:10.1090/tran/8054

On the dimension of subspaces of continuous functions attaining their maximum finitely many times
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by L. Bernal-González, H. J. Cabana-Méndez, G. A. Muñoz-Fernández and J. B. Seoane-Sepúlveda PDF

Trans. Amer. Math. Soc. 373 (2020), 3063-3083 Request permission

Abstract:

If $V$ stands for a subspace of $\mathcal {C}(\mathbb {R})$ such that every nonzero function in $V$ attains its maximum at one (and only one) point, then we prove that $\mathrm {dim}(V) \le 2$. This provides the final answer to a lineability problem posed by Vladimir I. Gurariy in 2003. Moreover, we generalize the previous result in the following terms: If $m \in \mathbb {N}$ and $V_m$ stands for a subspace of $\mathcal {C}(\mathbb {R})$ such that every nonzero function in $V_m$ attains its maximum at $m$ (and only $m$) points, then $\mathrm {dim}(V_m)\le 2$ for $m > 1$ as well. Besides being a problem closely related to real analysis, this problem actually needs the use of tools from general topology, geometry, and complex analysis, such as decompositions (or partitions) of manifolds or Moore’s theorem, among others.

References

Richard M. Aron, Luis Bernal González, Daniel M. Pellegrino, and Juan B. Seoane Sepúlveda, Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3445906
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
L. Bernal-González, G. A. Muñoz-Fernández, D. L. Rodríguez-Vidanes, and J. B. Seoane-Sepúlveda, Algebraic genericity within the class of sup-measurable functions, J. Math. Anal. Appl. 483 (2020), no. 1, 123576, 16. MR 4021906, DOI 10.1016/j.jmaa.2019.123576
Luis Bernal-González, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71–130. MR 3119823, DOI 10.1090/S0273-0979-2013-01421-6
G. Botelho, V. V. Fávaro, D. Pellegrino, J. B. Seoane-Sepúlveda, and D. Cariello, On very non-linear subsets of continuous functions, Q. J. Math. 65 (2014), no. 3, 841–850. MR 3261969, DOI 10.1093/qmath/hat043
D. Cariello, Analytical techniques on multilinear problems, Thesis (Ph.D.)–Universidad Complutense de Madrid, 2016.
Daniel Cariello and Juan B. Seoane-Sepúlveda, Basic sequences and spaceability in $\ell _p$ spaces, J. Funct. Anal. 266 (2014), no. 6, 3797–3814. MR 3165243, DOI 10.1016/j.jfa.2013.12.011
Krzysztof C. Ciesielski and Juan B. Seoane-Sepúlveda, Differentiability versus continuity: restriction and extension theorems and monstrous examples, Bull. Amer. Math. Soc. (N.S.) 56 (2019), no. 2, 211–260. MR 3923344, DOI 10.1090/bull/1635
Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. MR 872468
Per H. Enflo, Vladimir I. Gurariy, and Juan B. Seoane-Sepúlveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc. 366 (2014), no. 2, 611–625. MR 3130310, DOI 10.1090/S0002-9947-2013-05747-9
José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, Víctor M. Sánchez, and Juan B. Seoane-Sepúlveda, Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3863–3876. MR 2679609, DOI 10.1090/S0002-9939-2010-10420-3
José L. Gámez, Gustavo A. Muñoz-Fernández, and Juan B. Seoane-Sepúlveda, Lineability and additivity in $\Bbb R^{\Bbb R}$, J. Math. Anal. Appl. 369 (2010), no. 1, 265–272. MR 2643865, DOI 10.1016/j.jmaa.2010.03.036
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
Jerrold E. Marsden and Michael J. Hoffman, Basic complex analysis, 2nd ed., W. H. Freeman and Company, New York, 1987. MR 913736
Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR 1650235, DOI 10.1007/978-1-4612-0603-3
R. L. Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), no. 4, 416–428. MR 1501320, DOI 10.1090/S0002-9947-1925-1501320-8
Juan B. Seoane, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Kent State University. MR 2709064