I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature.
I'm hoping there's a nice proof of the Closed Graph Theorem of the following form. Let $T:X \to Y$ be a closed linear map between Banach spaces. Define a family $\{T_{\alpha}\}_{\alpha \in A}$ of bounded linear maps from $X$ to $Y$ (or from $X$ to another normed space $Z$) such that $\sup_{\alpha \in A} \| T_{\alpha}(x) \| < \infty$ for all $x \in X$ and $\| T \| \leq \sup_{\alpha \in A} \| T_{\alpha}\|$. Conclude using Uniform Boundedness Principle. Of course, coming up with the family $\{T_{\alpha}\}_{\alpha \in A}$ is the hard part.
Similarly/Alternatively, I'd be very happy to see a proof of the Bounded Inverse Theorem of the following form. Let $T:X \to Y$ be a bounded linear bijection between Banach spaces. Define a family $\{S_{\alpha}\}_{\alpha \in A}$ of bounded linear maps from $Y$ to $X$ such that $\sup_{\alpha \in A} \| S_{\alpha}(y) \| < \infty$ for all $y \in Y$ and $\| T^{-1} \| \leq \sup_{\alpha \in A} \| S_{\alpha}\|$. Conclude using Uniform Boundedness Principle.
I was inspired by this proof of the Uniform Boundedness Principle from the Closed Graph Theorem: https://math.stackexchange.com/a/1473367/570438 It looks at the map $\Phi(x) = (T_{\alpha}(x))_{\alpha \in A}$, which maps $X$ to the space of bounded maps in $Y^A$.
A similar question was asked here before, but without a satisfactory answer: https://math.stackexchange.com/questions/1703248/does-the-closed-graph-theorem-follow-from-banach-steinhaus?noredirect=1#comment6033062_1703248
Theorem 27.26-27.31 of Schechter's Handbook of Analysis and its Foundations gives a indirect argument. Relatedly, the argument is adapted to give a direct proof of the Open Mapping Theorem from the Uniform Boundedness Principle here: Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
I am aware of the standard arguments for the implications Open Mapping Theorem $\Leftrightarrow$ Bounded Inverse Theorem $\Leftrightarrow$ Closed Graph Theorem.
I originally posted at MSE with upvotes but no responses, so I'm cross-posting here. MSE Link: https://math.stackexchange.com/questions/2924391/direct-proof-of-closed-graph-theorem-or-bounded-inverse-theorem-from-uniform-b
