Let $ X$, $ Y$, $ Z$, be Banach spaces and let $ T:X\to Y $ and $ S:Y\to Z $ be linear transformations.Suppose that $S$ is Bounded and injective and that $ S \circ T $ is bounded.Prove that $T $ is bounded.
So now, $ S \circ T :X \to Z $
take any $ x\in X $
Since $ S \circ T $ is bounded, $ \|S \circ T(x)\| \leqslant\|S \|\cdot\|T \|\cdot\|x \| $ and since $ S $ is bounded $\exists$ $ M>0 $ s.t $ \|S\|\leqslant M $.
How can I use these results to prove that $T$ is bounded? This is given as an application of closed graph theorem.So how can we use it to solve the problem.
Any help is appreciated!