Linear Functional: Continuous?

Question

Given a Banach space: $E$
and chosen a Hamel basis: $\mathcal{B}$

Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined linearly and extended linearly}$$ How to show that the induced linear functionals are continuous iff the Banach spaces is finite dimensional?

Answer 1

Suppose that $\mathcal B$ contains a sequence $(b_k)_{k\geqslant 1}$ such that $\lVert b_k\rVert=1$ for each $k$. Define $$L_n(x)=\sum_{k=1}^nk\cdot \delta_{b_k}(x).$$ Then for each $x$, $\sup_{n\geqslant 1}|L_n(x)|$ is finite. Since $\lVert L_n\rVert\geqslant n$, the principle of uniform boundedness implies a contradiction.