Following the conventions from Heil: "A Basis Theory Primer" and Albiac, Kalton: "Topics in Banach Space Theory", we might define a basis of an (infinite-dimensional) normed space $V$ as a sequence $(e_n)$ in $V$, such that for any $x \in V$ there is a unique sequence of scalars $(\lambda_n)$, such that $x = \sum_n \lambda_n e_n$ (converging in norm), whereas for a Schauder basis we demand that these coefficients are produced by linear continuous functionals $(e^*_n)$, such that $e^*_m(e_n) = \delta_{mn}$ and $\lambda_n = e^*_n(x)$.

Now, if we work in a separable Banach space, these two notions coincide (theorem 4.13 in Heil and theorem 1.1.3 in Albiac, Kalton), but what if $V$ is a separable normed space which is not complete? Is there a simple, instructive example in which these linear functionals $(e^*_n)$ exist but fail to be continuous?