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. 

