Can I specifically calculate norms of an element in Banach space with Schauder basis?

Question

$X$ is a Banach space with Schauder basis $\{x_i:i\in \mathbb{N}\}$. Pick an element $x\in X$.

Let $x=\Sigma_{i=1}^{\infty}\alpha_ix_i$ Can I calculate its norm with repect to $\{\alpha_i\}$?

I guess its $\Vert x \Vert=\sqrt{\Sigma\vert\alpha_i\vert^2\Vert x_i\Vert^2}$, but I'm not sure. Is it correct?

Answer 1

In general computing norms even of finitely-supported vectors in a Banach space is hard, and the Tsirelson space is a notable example. Of course, one may cheat and say that such formula does exist, here it is:

$$\|x\|=\sup_n \|\sum_{k=1}^n a_k e_k\|,$$

here $(e_k)_{k=1}^\infty$ is a Schauder basis for your space.