On the page 247 of the book An introduction to Banach space theory by Robert E. Megginson, there is a lemma to be used in the proof of the Eberlein-$\check{\text{S}}$mulian theorem:
2.8.5 Day's lemma Let X be a normed space.
(a) If A is a relatively weakly limit point compact subset of X and $x_0 \in \overline{A}^w$, then there is a sequence in A that converges weakly to $x_0$.
(b) If $A_*$ is a relatively weakly limit point compact subset of $X^*$ and $x_0^* \in \overline{A_*}^{w^*}$, then there is a sequence in A that converges weakly to $x_0^*$.
Here are pages from google books.
I have some questions concerning about the statement and some details in the proof.
(1) I believe that part (b) should be
If $A_*$ is a relatively weakly-star limit point compact subset of $X^*$ and $x_0^* \in \overline{A_*}^{w^*}$, then there is a sequence in A that converges weakly-star to $x_0^*$ ,
and correspondingly all of the "weak" convergences and "weak" closure in the proof should be replaced by "weak-star". Am I right?
(2) In the second paragraph of the page 248, I don't understand why it follows easily from $$\|x^*\|\le 2~\text{sup}\{|x^*x|:x\in D\}, \forall x^* \in \langle \{x_n^*:n\in \mathbb{N} \} \rangle $$
that
$$\|x^*\|\le 2~\text{sup}\{|x^*x|:x\in D\}, \forall x^* \in [ \{x_n^*:n\in \mathbb{N} \} ] $$
where $\langle S \rangle$ means the smallest subspace containing $S$ and $[ S ]$ means the smallest weakly-star closed subspace containing $S$.
For (2), is it legal to take a sequence in $\langle \{x_n^*:n\in \mathbb{N} \} \rangle$ weakly-star converging to $x^*$?(In general, we should take a net instead of a sequence.)
Thanks in advance!
The statement and proof in my book are correct as given. I have replied directly to the questioner with further information.