Difference between topology of pointwise convergence, weak and weak$^*$ topologies

Question

In the book Exercises in Functional Analysis by Costara and Popa, page $411$, question $37$ goes as follows:

Let $X$ be a Banach space and $E$ be a subspace of $X^*.$ Suppose that $E$ separates points of $X$ and $B_X$ (unit ball of $X)$ is compact in the topology of pointwise convergence on $E.$ Then $X$ is a dual of a Banach space ($X$ is isometrically isomorphic to $\overline{(sp(E))}^*$).

In the solution given, the authors give the following sentences.

Let $\tau_p$ be the topology of pointwise convergence on $E,$ that is, the topology generated by the family of seminorms $(p_{x^*})_{x^*\in X^*}$ where $p_{x^*}:X\to\mathbb{R}$ is defined by $p_{x^*} =|x^*(x)|$ for all $x\in X.$ We shall prove that $T:(B_X,\tau_p)\to (\overline{(sp(E))}^*, weak^*)$ is continuous. Let $x_0\in B_X$ and $(x_\delta)_{\delta\in\Delta}$ be a net in $B_X$ such that $x_\delta\to x_0$ in the $\tau_p$ topology. Equivalently, $x^*(x_\delta)\to x^*(x_0)$ for all $x^*\in E.$

Here are my questions:

($1$) How can we equipped $B_X$ with $\tau_p$ when $\tau_p$ is meant for subspace of its dual space? Is it because we can view $X$ as a subspace of $X^{**}?$

$(2)$ When $B_X$ is equipped with $\tau_p,$ it seems that convergence in $\tau_p$ topology is just weakly convergent, am I right?

$(3)$ In conclusion, when we equipped $\tau_p$ on a dual space, then convergence in the topology is weak$^*$ convergent while when equipped on a Banach space convergence in the topology is weak convergent? Is the statement correct?

Answer 1

1. The topology $\tau_p$ is explicitly defined as a topology on $E$. It is an example of a topology generated by a family of seminorms, the seminorms being $x\mapsto |x^*(x)|$ for $x^*\in X^*$.

2. No, it is the weak* topology on $E$. There is no weak topology on $E$ considered in this context. (This topology would arise if one considered some Banach space $B$ such that $B^*$ is isomorphic to $E$.)

3. Unclear, perhaps because of the aforementioned misconceptions. The topology of pointwise convergence comes from thinking of the elements of our space as functions. We can think of the elements of $E$ as functions on $E^*$, and get weak* topology on $E$. We can think of the elements of $E^*$ as functions on $E$, and get weak topology on $E^*$.