Bounded Right Inverse

Question

If a linear operator between two Banach spaces is surjective and bounded, can we get any information about a right inverse? For example, is it bounded?

Thanks, trying to understand trace operator stuff on the Sobolev spaces.

Answer 1

If $T: X \to Y$ is bounded and surjective, the Open Mapping Theorem says there is an isomorphism $S:\; X/\ker(T) \to Y$ such that $T = S \circ \pi$, where $\pi: X \to X/\ker(T)$ is the quotient map. The trouble is, a quotient of $X$ might not be isomorphic to a closed subspace of $X$, so there might not be a bounded right inverse. For example, every separable Banach space is isomorphic to a quotient of $\ell^1$, but not every Banach space is isomorphic to a subspace of $\ell^1$ (e.g. $\ell^2$ is not).