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