The problem of Schauder decomposition of a given Banach space seems to play an important role in the geometry of Banach spaces, especially when one is interested in finite dimensional Schauder decompositions (FDD).
I am wondering if the Schauder decomposition can be regarded (in special cases) as the internal counterpart to infinite sums of Banach spaces.
Let me consider two cases: $C(K)$ and $L^p(\mu)$.
Suppose that $E$ is either $C(K)$ space for some compactum $K$ or $L^p(\mu)$ for some measure $\mu$.
Let $(E_n)$ be a sequence of complemented copies subspaces of $E$ such that for each integer $n$ $$E_1\oplus \ldots \oplus E_n \cap E_{n+1}=\{0\}.$$
In the $C(K)$ case assume that each $E_n$ is isomorphic to $c_0$ and in the latter one, $E_n$ is isomorphic to $\ell^p$.
Define $F$ to be the closed linear span of all $E_n$. Is the family $$\{E_1\oplus \ldots \oplus E_n\colon n\in \mathbb{N}\}$$ a blocking Schauder decomposition for $F$?
Is $F$ isomorphic to $c_0$ / $\ell^p$ ?
