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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?

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Convince 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. –  t.b. Sep 11 '11 at 16:14
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Mackey 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 –  GEdgar Sep 11 '11 at 17:02
    
I think that GEdgar means: Kelley, Namioka, Linear topological spaces, Springer-Verlag, 1976. –  acgm Mar 5 at 16:04

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