I have been trying to understand the weak star topology, but as is often the case the material online is quite hard to understand.
For a Banach space $X$ and the dual space $X^{*}$ could someone please explain what the weak star topology on $X^{*}$ is?
The weak star topology on $X^\ast$ is the subspace topology we inherit when we see $X^\ast$ as a subspace of $\Bbb R^X$ (the set of all functions from $X$ to $\Bbb R$) (assuming the reals as underlying field here). $\Bbb R^X$ has the usual product topology (i.e. the minimal topology that makes all all projections (or really evaluations here) continuous). That's really all there is to it.