Note : This question was asked 7 hours ago but closed so I thought about it again and asking it with my attempts . Hope , it on -site now.
The following question was asked in a masters exam for which I am preparing and I was unable to solve it. So, I am asking for help here.
Both continuity and entire function implies that sup will be attained as set is compact. entire implies further that sup will be attained at the boundary. Also, $||f||_K$ =0 implies that f(z)=0.But By this I marked all options are correct but answer is
D only
So, Please tell what mistake I am making . I have done a course on functional analysis and followed Krieszig.
Can you please give outline of solution?
$\Vert f\Vert_K=0$ does not imply that $f=0$. First, it is possible that $K=\{z_0\}$, a single point. For example, if $K=\{0\}$ and $f(z)=z^2$, then $\sup_{z\in K}|f(z)|=0$, so $\Vert f\Vert_K=0$, but $f\neq 0$. This invalidates options A and B. If $K$ has non-empty interior, then it is still possible that a continuous function $f$ is zero on all of $K$, but nonzero off of $K$. For example, $K=\{z\in\mathbb C:|z|\leq 1\}$ and $$f(z)=\begin{cases} 0 & |z|\leq 1\\ |z|-1 & |z|>1\end{cases}.$$ This invalidates C. But if an entire function is zero on a nonempty open set (such as the interior of $K$), then it is zero on all of $\mathbb C$. Therefore, $D$ is correct.