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?