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Let $X$ be a set in a $\mathbb{C}$-Banach-Space. Given is a sequence $(c_n)_{n\in N}$ so that the power series $$\left(\sum_{n=0}^N c_n(z-a)^n)\right)$$ has a radius of convergence $\rho \gt 0$. Further given is a function $$f:{z\in C:|z-a|\lt\rho} \to X$$ assigning $$z \to \sum_{n=0}^\infty c_n(z-a)^n$$ Let further be $(y_m)_{n\in N}$ be a sequence in $({z\in C:|z-a|\lt\rho})$ with $a=\lim_{m \to \infty}y_m$ and $f(y_m)=0$ for all $m\in N$.

Show that $c_n=0$ for all $n \in N_0$.

I do not really know what to do there and would appreciate a hint how to start. Thank you very much

Netchaiev
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Michael Maier
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  • If $a=\lim_{m\rightarrow}y_m$, as I assumed, it is just a consequence that the zeros of an holomorphic non-zero function (here $f$) must be isolated. – Netchaiev Jan 24 '18 at 09:37
  • Thank you. I am sorry, a is defined as $a=\lim_{m\to\infty}(y_m)$ However, I do not quite get what you mean;( – Michael Maier Jan 24 '18 at 09:46

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Since $f$ is holomorphic on $D(a,\rho)$, if $\lim_{m\rightarrow +\infty} y_m=0$, we have $f(a)=0$ (since $f$ is indeed continuous). But it implies that $a$ is a zero of $f$ and he is not isolated (since $f(y_m)=0$) : $f$ being holomorphic, $f$ must be equal to zero on $D(a,\rho)$. The very same isolated zeros principle, implies on its turn that every $c_n=0$.

Netchaiev
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