I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$.
I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb N}\bigcup\{−e_n\}_{n \in \mathbb N}$, and for $\mathcal{C}[0,1]$ the constant functions $f=1$ and $f=−1$).
I had a little trouble figuring out what should be the answer in case the field is $\mathbb{C}$. I think it should be (for $\ell^1$) all the $\{e_n\}_{n\in \mathbb N}\bigcup\{−e_n\}_{n \in \mathbb N}\bigcup\{ie_n\}_{n\in \mathbb N}\bigcup\{−ie_n\}_{n \in \mathbb N}$, and (for $\mathcal{C}[0,1]$) the functions $f=1,f=−1,f=i,f=−i$. But I can also see it go to (for $\ell^1$) all the $\{ae_n:|a|=1\}$, and (for $\mathcal{C}[0,1]$ all the constant functions $f=a$ with $|a|=1$.
I would like some guidness if possible. Knowing the right answer could help, as well as the intuition behind and maybe some tips on how to get there. Thanks!!