I've just thought about this question. Could you elaborate on it?
Let $E$ be an infinite-dimensional t.v.s. Let $B$ be a basis of $E$, i.e., each $x\in E$ is a finite linear combination of some elements in $B$ and each finite subset of $B$ is linearly independent.
For each $x \in E$, there is a unique collection $(\lambda_{x,b})_{b\in B}$ such that $$x = \sum_{b\in B}\lambda_{x,b} b.$$ By definition, $(\lambda_{x,b})_{b\in B}$ has finitely many non-zero terms. For each $b \in B$, we define $f_b:E \to \mathbb R$ by $f_b(x) := \lambda_{x,b}$. Clearly, $f_b$ is linear.
Is $f_b$ continuous?
