Existence of right inverse.

Question

We know that a surjective continuous linear map $ T : X \to Y$ has a right inverse iff $ \ker(T)$ is complemented. Here $X$ and $Y$ are Banach spaces. Is this result true for locally convex topological vector spaces or even for Frechet spaces ? If not, are there any analogues in these generalisations ? Any light on this would be extremely helpful. Thank you.

Answer 1

Let's take a look at the proof of the Banach space statement you cite. As we will see it easiliy generalizes to Frechet spaces

If $T \colon X \to Y$ has a right inverse, say $S \colon Y \to X$, then $TS = \def\Id{\mathord{\rm Id}}\Id_Y$. Let $P := ST$. Then we have $P^2 =(ST)^2 = S(TS)T = ST=P$, so $P$ is a continuous projection. If $x \in \ker P$, then $STx = 0$, hence $Tx = TSTx = 0$, so $x \in \ker T$, as $\ker T \subseteq \ker P$ trivially, $\ker T = \ker P$. Hence $\Id - P$ is a continuous projection onto $\ker T$ and hence $\ker T$ is complemented.

If $\ker T$ is complemented, say $P \colon X \to X$ is a continuous projection with $\ker P = \ker T$, then $\mathop{\rm im} P$ is a closed subspace and $T' := T|_{\mathop{\rm im} P}$ is continuous, one-to-one and onto. Hence, by the open mapping theorem (as a consequence of the Baire category theorem, it is true for Frechet spaces), $T'$ is invertible, set $S' := (T')^{-1} \colon Y \to \mathop{\rm im} P$. Define $S \colon Y \to X$ by $Sy = S'y$, then $TS = \Id_Y$.