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

Theorem

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

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

Let $\struct {Y, \tau_Y}$ be a topological vector space over $\GF$.

Let $T : X \to Y$ be a continuous linear transformation such that:

$T \sqbrk X$ is non-meager in $\struct {Y, \tau}$.

Then:

$(1): \quad$ $T \sqbrk X = Y$

$(2): \quad$ $T$ is open

$(3): \quad$ $\struct {Y, \tau_Y}$ is an $F$-space

Corollary

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.