Let $X$ be a Banach space and $f_n$ be a sequence in the dual space $X^*$ such that for all $x \in X$, the sequence $f_n(x)$ converges. Show that $(f_n)$ exists a weak-* limit $f \in X^*$.
In Kreyszig's book, the above statement is simply a definition. I do not know what to prove and how to prove. Could anyone help me, please? Thank you!
The assumption is that for every $x\in X$, the limit of $f_n(x)$ exists. That defines a map $f\colon X \to \mathbb{C}$ (or $\mathbb{R}$) as the pointwise limit. What needs to be shown is that $f\in X^\ast$, i.e. that $f$ is linear and continuous. The linearity is direct, and for the continuity, the Banach-Steinhaus theorem can be useful.
Note that the completeness of $X$ is important, for merely normed spaces, the pointwise limit need not be continuous.
I don't know. If Kreyszig states that $f_n$ is weak$^\ast$ convergent to $f\in X^\ast$ when $f_n(x) \to f(x)$ for all $x$, then that is the usual definition. But if Kreyszig defines $(f_n)$ to be weak$^\ast$-convergent if for every $x\in X$ the sequence $(f_n(x))$ converges to something, then that coincides with the usual definition for Banach spaces, but would be incorrect for incomplete normed spaces. I expect that the existence of $f\in X^\ast$ such that $f_n(x) \to f(x)$ is assumed somewhere around there.