Given that $C$ is a subspace of $l_\infty$ consisting of all convergent sequences. What is the general form of a bounded linear functional on C?
ok, so I know that a linear functional is a mapping from C to field such that $f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$ and it will be bounded if there exists some $N>0$ such that $\|f(x)\|\leq N\|x\|$ for all $x \in C$. Now how do I generalize $f$?
Assume that $f\in C^{\ast}$, then let $\beta_{n}=f(e_{n})$, where $e_{n}$ is the sequence of $i$th entry $1$ and zero otherwise. Now let $\gamma_{n}$ be such that $|\gamma_{n}|=1$ and that $\gamma_{n}\beta_{n}=|\beta_{n}|$, then \begin{align*} \sum_{n=1}^{m}|\beta_{n}|=\sum_{n=1}^{m}\gamma_{n}\beta_{n}=f\left(\sum_{n=1}^{m}\gamma_{n}e_{n}\right)\leq\|f\|\left\|\sum_{n=1}^{m}\gamma_{n}e_{n}\right\|_{\infty}=\|f\|, \end{align*} taking $m\rightarrow\infty$, we have $(\beta_{n})\in l^{1}$. Now \begin{align*} f(x_{n})=f\left(\sum_{n=1}^{\infty}x_{n}e_{n}\right)=\sum_{n=1}^{\infty}f(x_{n}e_{n})=\sum_{n=1}^{\infty}x_{n}f(e_{n})=\sum_{n=1}^{\infty}x_{n}\beta_{n}. \end{align*}
