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Non-continuous linear functionals on topological vector spaces. (English) Zbl 1134.46002

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.

MSC:

46A03 General theory of locally convex spaces