One often considers a Banach space $X$ under the "weak topology", ie. the weakest topology such that all bounded linear functionals are continuous. This leads me to wonder about the weakest topology on $X$ such that ALL linear functionals (including unbounded ones) are continuous. A priori this topology is not necessarily stronger or weaker than the standard norm topology. Does it possess any interesting properties? Is it Hausdorf? Connected? Discrete, even?
Weakest topology with respect to which ALL linear functionals are continuous
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general-topology
functional-analysis
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0I think that GEdgar means: Kelley, Namioka, Linear topological spaces, Springer-Verlag, 1976. – 2014-03-05