Let $\mathcal{A}:X\to Y$ be continuous linear operator, $X$ and $Y$ are Banach spaces. Let $\text{Im} \mathcal{A}=Y$.
Is $\ker\mathcal{A}$ a complemented subspace of $X$?
Let $\mathcal{A}:X\to Y$ be continuous linear operator, $X$ and $Y$ are Banach spaces. Let $\text{Im} \mathcal{A}=Y$.
Is $\ker\mathcal{A}$ a complemented subspace of $X$?
No, not in general. For example, if $X$ is not isomorphic to a Hilbert space then $X$ contains a noncomplemented subspace $Z$. The quotient map $X \longrightarrow X/Z$ is surjective, continuous and linear, but the kernel (which is $Z$ of course) is not complemented.