# Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle

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: Does the Closed Graph Theorem follow from Banach-Steinhaus?

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: https://mathoverflow.net/questions/190587/is-there-a-simple-direct-proof-of-the-open-mapping-theorem-from-the-uniform-boun

I am aware of the standard arguments for the implications Open Mapping Theorem $$\Leftrightarrow$$ Bounded Inverse Theorem $$\Leftrightarrow$$ Closed Graph Theorem.