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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4Convince yourself that *all* semi-norms must be continuous. In particular the topology is stronger than the norm topology. Connected, yes, every topological vector space is connected, as scalar multiplication is continuous. Discrete? No unless $X=0$, as scalar multiplication must be continuous. Interesting property? All linear maps into a locally convex topological vector space are continuous. – 2011-09-11
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3Mackey studied "weak" topologies $\sigma(X,Y)$ where $Y$ is any subspace of $X' = $ the set of all linear functionals on $X$. In particular $\sigma(X,X')$ is the topology of your quesion. Find a lot more in Kelly--Namioka *Topological Vector Spaces* – 2011-09-11
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0I think that GEdgar means: Kelley, Namioka, Linear topological spaces, Springer-Verlag, 1976. – 2014-03-05