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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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up vote 2 down vote accepted

No. Consider $F(x)=(f(x),0)$ where $f$ does not have closed range.

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I felt sure there was a simple counter-example to the question as stated, but was trying with the identity as the other factor. D'oh! –  Andrew Stacey Jul 29 '10 at 18:40
    
Thanks much, Bill. It is embarassing not to have considered mapping into one of the factors (I should have known better). –  Chris Leary Jul 29 '10 at 19:03
    
Does the answer change if the linear maps $F(i)$ must both be nonzero? –  Chris Leary Jul 29 '10 at 19:41
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No; make the mapping into into the second factor rank one instead of rank zero. –  Bill Johnson Jul 29 '10 at 21:19
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