# Closedness of the image of the unit ball

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?

• We’re talking closed unit ball, I assume? – Anthony Quas Apr 15 at 15:47

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 $$(-2, 2)$$.