(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does one of these imply the other? (a) $X^\ast$ is weak* separable (b) $B_{X^\ast}$ is weak* s...

(b) implies (a): Let $D$ be a countable dense set in $B_{X^\ast}$. Then $\overline{\bigcup_{n=1}^\infty nD} \supseteq \bigcup_{n=1}^\infty nB_{X^\ast} = X^\ast$ and $\bigcup_{n=1}^\infty nD$ is countable. (a) does not imply (b): provided the preprint Antonio Avilés, Grzegorz Plebanek, José Rod...

Let $D$ be an open connected set in $\mathbb C$ and $\{f_n \}$ be a sequence of holomorphic functions in $D$ such that $f_n \to f$ uniformly on compact subsets of $D$ . If $f$ is non-constant and $z \in D$ then how to show that there exist a sequence $\{z_n\}$ in $D$ and positive integer $N$ suc...

I am working through some of the past qualifying exams in complex analysis and I am a bit stuck on the question I posed in the title. My immediately thought is use Rouche's Theorem. For instance, I tried letting $f(z)=e^{z}$ and $g(z)=3z+1$ in hopes of getting $|f(z)|\leq |g(z)|$ on $|z|=1$. But ...

First, some notations and definitions: 1) For a vector space $V$:$$\mathsf{End}(V):=\{\mathsf{Linear}\ \mathsf{maps}\ f:V\longrightarrow V\}.$$ 2) A linear representation of a group $G$ is a pair $(V, \rho)$ consisting of a vector space $V$ and a map $\rho:G\longrightarrow \mathsf{End}(V)$ such ...