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** 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 $c\_0$ / $\ell^p$ ?



Recovering Schauder decompositions
No. You need the projections $Q\_n$ onto $E\_1\oplus \dots E\_n$ from $F$ to be uniformly bounded in order for $(E\_n)$ to be a Schauder decomposition for $F$. Even then $F$ need not be isomorphic to $c\_0$/$\ell\_p$. However, if the $Q\_n$ are uniformly bounded from $\ell\_p$, then by taking limits in the weak operator topology you get (when $1<p<\infty$) a projection from $\ell\_p$ onto $F$ and hence $F$ is isomorphic to $\ell\_p$. That is not the case for $p=1$ or in the $C(K)$ case. It is true in the $c\_0$ case, because you get by using the weak\* operator topology in $\ell\_\infty$ an operator from $c\_0$ into $F^{\*\*}\subset \ell\_\infty$ that is the identity on $F$, which implies that $F$ is a $\mathcal{L}\_\infty$ space, and every $\mathcal{L}\_\infty$ subspace of $c\_0$ is isomorphic to $c\_0$. 


If you do not understand, email me questions and I will explain when I have more time.


