I am given a functional $f$, that is defined as \begin{align} f:c_0 &\rightarrow \mathbb{R} \\ x &\mapsto \sum_{k=1}^\infty \frac{\xi_k}{3^k} ~~~x = (\xi_k)_{k\ge1} \end{align} and I am supposed to calculate ||f||.
I started by estimating an upper bound for $||fx||$:
$$ ||fx|| = \left| \sum_{k=1}^{\infty} \frac{\xi_k}{3^k} \right| \le \sum_{k=1}^{\infty} \left| \frac{\xi_k}{3^k} \right| \le \sum_{k=1}^{\infty} \frac{\sup_\limits{k \ge 1} |\xi_k|}{3^k} \le ||x|| \sum_{k=1}^{\infty} 3^{-k} \le \frac{1}{2} ||x|| $$
This tells me that $||f|| \le \frac{1}{2}$. So now I'd like to show that there exists an $x$ such that $||f|| \ge \frac{1}{2}$, but I can't find a suitable $x$ to do the job. Am I approaching this problem the wrong way or is there some obvious $x$ that I'm somehow not seeing?
you can take the sequences $a^n = (1,1,\underbrace{...}_{n\text{ times}},1,0,0,...)$