TITLE: Surjective bounded operator in Banach spaces without bounded right-inverse
QUESTION [10 upvotes]: Could someone give a simple example of surjective (not bijective) bounded operator in Banach spaces without bounded right-inverse?

REPLY [2 votes]: Let's take $c_o$. It's uncomplemented subspace of $l_\infty$.  Let $T: l_\infty \to l_\infty/c_0$. Both spaces are Banach. T is bounded and surjective (not bijective). According to theorem from Existence of right inverse. if $T$ has a right inverse, $ker~T$, which is $c_0$ in our case, should be complemented, but it's not true. Hence, $T$ has no right inverse at all.
Actually  we can take $T: X\to X/Y$, where $X$ is Banach space and $Y$ is closed uncomplemented subspace of X.