# Uniform boundedness principle and closed graph Theorem

In a functional analysis class, my professor gave a problem to prove the uniform boundedness principle from the Closed graph theorem.

The problem goes thus:

Let $$(X,\|\cdot\|_X)$$ and $$(Y,\|\cdot\|_Y)$$ be real Banach spaces. Let $$\{T_{\alpha}\}_{\alpha \in \Delta}$$ be a collection of bounded linear maps from $$X$$ to $$Y$$. Consider a new norm defined on $$X$$ $$\|x\|^{'} :=\|x\|_{X} + \sup_{\alpha \in \Delta}\| T_{\alpha} x\|_{Y}$$ where $$\Delta$$ is an arbitrary index set . Deduce from the closed graph theorem the uniform boundedness principle.

I have the following questions :

1. The standard proof of the uniform boundedness principle uses the Baires category theorem which I am very well conversant with. In that proof, we only require that $$X$$ be Banach contrary to the assumption of the closed graph theorem where both $$X$$ and $$Y$$ should be Banach. Can someone explain how the closed graph theorem and the uniform boundedness theorem would then relate.

2. What function can one define on $$X$$ with the new norm that would enable me apply the closed graph theorem. I am basically new to the course and would like to know what would motivate one to define such function.

Though I have searched online and saw this. I understand the proof but it has no affiliation with the new norm as given by my problem. Thanks

1. Suppose that for every $$x\in X, \sup_{\alpha\in \Delta}\|T(x)\|_Y$$ is bounded, show that $$(X,\|\cdot\|')$$ is Banach.
2. Consider $$Id_X:(X,\|\cdot\|)\rightarrow (X,\|\cdot\|')$$. Its graph is closed, so it is a bounded map. You can deduce that there exists $$C>0$$ such that $$\|x\|_X+ \sup_{\alpha\in \Delta}\|T_{\alpha}(x)\|_Y.