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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:
46A03General theory of locally convex spaces