# Understanding topologies in dual and bidual.

I don't know if this is a dumb question but i think i better ask and get my confusion clarified .

Talking about topologies in a Vector space , Topologies induced by norms are pretty easy to understand. The open sets of the topology are the open balls with respect to the norm .

But now its not clear to me how the topology in the dual space will be like , how do the open balls or sets will look like for the continuous functionals in the dual ? and same with the bidual ?

We generally define a norm on the functional , and we define openness using this norm right ? I am sort of not able to imagine or get get the feeling about weak topology or weak star topology .

I would like your suggestions , explanations, recommendation. I really need to understand these things clearly as soon as possible. Thank you for your help. Merry Christmas!

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One way to get a handle/feeling on the topology is to see what it means for a sequence to converge in the topology. For norm topology on the dual space, $f_n \to f$ iff $\|f_n - f \| \to 0$, which basically means $f_n$ converges to $f$ uniformly on the unit sphere. For weak topology, $f_n \to f$ iff it converges pointwisely. – user27126 Nov 30 '12 at 23:03
Things are also made more complicated by the fact that, depending on the topology you take on the dual space, the dual of the dual is different. – Christopher A. Wong Dec 1 '12 at 0:28