I have a Banach space $B$ and a continuous linear transformation $F:B \rightarrow B\times B$. One of the induced transformations $F(1):B \rightarrow B$ and $F(2):B \rightarrow B$ into the factors of the product has closed range. Must F have closed range? I have the max norm on the product, i.e., $\F(x) \ = max\{\F(1)(x)\, \F(2)(x)\\}$ for $x$ in $B$. I was hoping to use the minimum moduli of the $F(i)$ to provide an affirmative answer.
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No. Consider $F(x)=(f(x),0)$ where $f$ does not have closed range.

$\begingroup$ I felt sure there was a simple counterexample to the question as stated, but was trying with the identity as the other factor. D'oh! $\endgroup$ – Andrew Stacey Jul 29 '10 at 18:40

$\begingroup$ Thanks much, Bill. It is embarassing not to have considered mapping into one of the factors (I should have known better). $\endgroup$ – Chris Leary Jul 29 '10 at 19:03

$\begingroup$ Does the answer change if the linear maps $F(i)$ must both be nonzero? $\endgroup$ – Chris Leary Jul 29 '10 at 19:41

1$\begingroup$ No; make the mapping into into the second factor rank one instead of rank zero. $\endgroup$ – Bill Johnson Jul 29 '10 at 21:19