It is know that in a Hilbert space $H$, an orthonormal basis $(e_i)_{i\in \mathbb{N}}$ is weakly 2-summable, that is, $\sup_{h\in B_H}\sum_{i=1}^\infty \langle h,e_i\rangle^2<\infty$.
I was wondering if we have an analogous statement in a Banach space $B$ with a Schauder basis $(b_i)_{i\in \mathbb{N}}$. More precisely, is it true for some $p>0$ that $\sup_{b^*\in B_{B^*}}\sum_{i=1}^\infty \vert\langle b_i,b^*\rangle\vert^p<\infty$?