Derive the open mapping theorem from the inverse mapping theorem in the case $X$ is a Hilbert space.
Proof.
$X$ is a Hilbert space, $Y$ is a Banach space, $T\colon X→Y$ is a surjective linear map.
By Theorem 3.11, $TX$ closed implies $T^*Y$ closed.
Let $Q\colon X→T^*Y$ be the orthogonal projection.
$Q$ is open: $B^{T^*Y}=Q(B^{T^*Y})⊆Q(B^X)$.
Let $S≔T|_{T^*Y}$.
$S\colon T^*Y→Y$ is injective: if $x∈\ker(S)$ then $x∈\ker(T)∩T^*Y=\{0\}$.
By the inverse mapping theorem $S$ has a continuous inverse, so $S$ is open.
Since $\ker(T)^⟂=T^*Y$, for all $x∈X,T(x-Qx)=0⇒Tx=TQx$, so $T=S∘Q$ is open.
% https://math.stackexchange.com/questions/4330870/inverse-mapping-theorem-implies-open-mapping-theorem
[In general: Quotient map is surjective so open. Projection map is surjective so open.]