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

2019; American Mathematical Society; Volume: 373; Issue: 5 Linguagem: Inglês

10.1090/tran/8054

1088-6850

Luis Bernal–González, H.J. Cabana-Méndez, Gustavo A. Muñoz-Fernández, Juan B. Seoane‐Sepúlveda,

Mathematical Dynamics and Fractals

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.