Convergence of partial sums of basis vectors in banach space

Question

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i e_i$, is it true, that $p_n$ convergence is equivalent to $||p_n||$ convergence?

Answer 1

No. Let $B=c_0$ and let $(e_n)$ be the standard basis. Let $\alpha_j=1$. Then $||p_n||=1$ but $p_n$ does not converge.