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Let $X$ be a Banach space and let $P$ be a bounded, linear projection on $X$. Is $P[B_X]$ closed in $X$? Here $B_X$ is the closed unit ball of $X$.

This is trivial if $X$ is reflexive, but otherwise is it true?

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  • $\begingroup$ We’re talking closed unit ball, I assume? $\endgroup$ – Anthony Quas Apr 15 at 15:47
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Define $P: c_0 \to c_0$ by $P(a_0, a_1, a_2, \ldots) = (\sum \frac{a_n}{2^n}, 0, 0, \ldots)$. Then $P(a_0, 0, 0, \ldots) = (a_0, 0, 0, \ldots)$, so this is a projection onto the first coordinate. But the image of the closed unit ball of $c_0$ under this map is the open interval $(-2, 2)$.

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