The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into subspaces of dimension $2$, and also admit a Schauder basis which is the union of some natural bases of the $2$-dimensional subspaces.
QUESTION: Suppose that, for some $k\in\mathbb N$, the Banach space $X$ admits a symmetric FDD into subspaces of dimension $k$.
Can we assure that $X$ admits a Schauder basis?
Best Answer
Yes. If $(E_n)$ is a FDD for $X$ where each $E_n$ has dimension $k$, then we can pick a basis $(e_i^n)_{i=1}^k$ for each $E_n$ with basis constant at most $\sqrt{k}$. Then the concatenation of $(e_i^n)_{i,n}$ in natural order is a Schauder basis for $X$ whose basis constant is less than or equal to $\sqrt{k}C$ where $C$ is FDD constant. The symmetry is not needed.