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Oct 18, 2015 at 17:46 comment added Pietro Majer @Jean Duchon: I think people here prefer not to follow up elementary or trivial questions, in order to keep the topics within research level. Answering in comments is intended to stop the activity showing that the question is indeed trivial.
Oct 18, 2015 at 10:13 comment added Jean Duchon @Christian Remling That's an answer, isn't it? (albeit to a trivial question). Why post answers as comments??
Oct 18, 2015 at 4:42 comment added Christian Remling $\ell^{\infty}[0,1]$ as described by the OP looks to me like bounded functions on $[0,1]$, with sup norm. So $x\mapsto x(s)$ is in the dual, and $f$ is weakly discontinuous everywhere.
Oct 18, 2015 at 4:19 comment added user81632 Please clarify the Riemann Integrability of a vector valued function. Hope $\lambda$ is the Lebesgue measure.
Oct 18, 2015 at 4:00 history answered Tanmoy Paul CC BY-SA 3.0