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?



Closedness of the image of the unit ball
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 $(-2, 2)$.


