User:Caliburn/s/fa/Banach-Schauder Theorem/F-Space/Corollary

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, d_X}$ be an $F$-space over $\GF$.

Let $\struct {Y, d_Y}$ be an $F$-space over $\GF$.

Let $T : X \to Y$ be a continuous surjective linear transformation.

Then $T$ is open.

Proof

From the Baire Category Theorem, $\struct {Y, d_Y}$ is a Baire space.

From Baire Space is Non-Meager, $\struct {Y, d_Y}$ is non-meager.

Since $T \sqbrk X = Y$, $T \sqbrk X$ is then non-meager.

Applying Banach-Schauder Theorem: $F$-Space, we obtain that $T$ is open.