Let $\{\varphi_j\}_{j=1}^{\infty}$ be a family of continuous linear functionals over a Banach space $X$ s.t. $\|x\|\leq \sum\limits_{j=1}^{\infty}|\varphi_j(x)|<\infty$. Prove that for every continuous linear functional $\varphi : X \longrightarrow \mathbb{K}$ there exists a bounded sequence $\{a_j\}_{j=1}^{\infty} \subset \mathbb{K}$ such that $\varphi(x) = \sum\limits_{j=1}^{\infty}a_j\varphi_j(x)$ for all $x\in X$.
It is obvious that $|\varphi(x)|\leq \alpha\|x\| \leq \alpha \sum_{j=1}^{\infty}|\varphi_j(x)|$ for some $\alpha$. It seems to me that it may be possible that this $\alpha$ is the upper bound of the desired sequence but I am not sure how to show it.
By assumption, $T:X\to\ell_1$, $x\mapsto (\varphi_n(x))_{n\in\mathbb N}$ is a well-defined map which is linear and has closed graph (due to the continuity of all $\varphi_n$). For $\varphi\in X^*$ define a functional $\psi: T(X) \to \mathbb K$ by $T(x) \mapsto \varphi(x)$. Then check that $\psi$ is well-defined and continuous with respect to the $\ell_1$-norm. By Hahn-Banach, you can extend $\psi$ to an element $\Psi \in \ell_1^*$ which has a representation $\Psi(y)=\sum\limits_{j=1}^\infty a_j y_j$ for a bounded sequence $a$. Then you get $$\varphi(x)=\psi(T(x))=\Psi(T(x))=\sum_{j=1}^\infty a_j\varphi_j(x).$$