Goldstein's theorem

Question

I recently picked up functional analysis and I'm still new to both $\omega$ and $\omega*$ topologies. To be precise, what is the topolgy on the dual of the dual of X (denoted by X**). Let $\pi:X\rightarrow X^{**}$ be the canonical embedding. The Goldstein's theorem states that $\overline{B_{\pi(X)}}^{\omega^*}=B_{X^{**}}$. Does that mean that we can define the $\omega*$-topology on $X^{**}$? Aren't the funcionals defined on $X^{**}$ containing the functionals defined on $X$ (If we concider that X is a subset of X**)?

Answer 1

Let space $Y$ be a Banach space that has a predual predual space $Z$, i.e. $Z^{\ast} =Y$. Then we may define the weak star topology on $Y$, as the coarsest topology that makes all $f(y)=<z,y>$ continuous.

In your context, the predual space of $X^{\ast\ast }$, is $X^{\ast}$. Therefore, the weak star topology on $X^{\ast\ast }$ is the coarsest one that makes all $f(x^{\ast \ast})=<x^{\ast},x^{\ast\ast}>$ continuous. This shouldn't be confused with the weak star topology on $X^{\ast}$, i.e. the coarsest topology on $X^{\ast}$ that makes all $f(x^{\ast})= <x,x^{\ast}>$ continuous.

Via your cannonical emebedding $\pi$, every linear functional $f$ on $X^{\ast \ast}$, induces a linear functional on $X$, namely $f\circ \pi$.