Relation between weak * topology of $X^*$ and relative norm topology

Question

Let $X$ be a normed space.

what is the relation between the weak* topology of $X^*$ and the topology of $X^*$ inherited from $BC(X,\mathbb C)$ (space of all bounded, continuous, complex valued functions with supremum norm $\|f\|=\sup_{x\in X}|f(x)|$) ?

Answer 1

The weak* topology is the coarsest/weakest topology on $X^*$ that makes the elements $X$ into continuous functionals on $X^*$ via: $$x:X^*\to\Bbb C, \ x^*\mapsto x^*(x)$$ It is easy to see that the norm topology also makes the elements of $X$ continous functionals on $X^*$, so the norm topology is finer/stronger than the weak* topology. But they are never equal unless $X$ is finite dimensional.