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Zbl 1134.46002
Garc\'ia-Pacheco, Francisco J.
Non-continuous linear functionals on topological vector spaces. (English)
[J] Banach J. Math. Anal. 2, No. 1, 11-15, electronic only (2008). ISSN 1735-8787/e

Let $X$ be a topological vector space. It is proved that if $X$ has a Hamel basis that is not closed, then there exists a totally discontinuous linear functional on $X$, and the space $X$ then possesses a convex balanced absorbing subset with empty interior. Conversely, if all convex, balanced, absorbing subsets of $X$ have nonempty interiors, then all linear functionals on $X$ are continuous (and thus $X$ is Hausdorff). The open question of the finite-dimensionality of $X$ in the above setting remains.
[Wlodzimierz Ślȩzak (Bydgoszcz)]

MSC 2000:: *46A03 General theory of locally convex spaces

Keywords: topological vector space; noncontinuous linear functional; convex set; balanced set; absorbing set

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