# Closed range for a continuous linear transformation

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