Let $X$ and $Y$ be Banach spaces that are known to have Schauder bases.
If $x_i$ is a Schauder basis for $X$, and $T:X \to Y$ is a linear homeomorphism, is it true that $Tx_i$ is a Schauder basis for $Y$?
It seems obvious that the answer is yes but I am only confused about the role of "unconditional convergence" here...