Let $X$ be a normed space and let $(x_n)$ be a sequence in $X$ such that $\sum\limits_{n=1}^{\infty}x^*x_n$ is convergent. I want to show that $f(x^*)=\sum\limits_{n=1}^{\infty}x^*x_n$ is a bounded linear functional.
Linearity is easy to show. For boundedness, $|f(x_n)|=|\sum\limits_{n=1}^{\infty}x^*x_n |\leq \sum\limits_{n=1}^{\infty}|x^*x_n|$. I got stuck here. Now I have no idea how to proceed. Please give a hint.
This is an immediate application of Uniform Boundedness Principle (also known as Banach - Steinhas Theorem). Define $T_n:x^{*} \to \mathbb R$ by $T_n(x^{*})=\sum_{k=1}^{n} x^{*}(x_k)$. It is easy to check that $\|T_n\| \leq \sum_{k=1}^{n}\|x_k\|$. By Uniform Boundedness Principle it follows that $\|T_n\|$ is bounded. Thus there is a finite consatnt $C$ such that $|\sum_{k=1}^{n} x^{*}(x_k)| \leq C\|x^{*}\|$ for all $x^{*}$. Letting $n \to \infty$ we get $|f(x^{*})| \leq C \|x^{*}\|$ so $f$ is continuous.