The Kalton-Peck Banach space $Z\_2$ (see Section 6 in [this paper](https://www.ams.org/journals/tran/1979-255-00/S0002-9947-1979-0542869-X/S0002-9947-1979-0542869-X.pdf)) 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?



Schauder bases in Banach spaces with a symmetric $k$-FDD
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.


