I am trying to prove that for normed vector spaces $U$, $V$, and $W$ with $S : U \rightarrow V$ and $T : V \rightarrow W$ linear maps. We have $||T \circ S|| \leq || T|| \cdot ||S||$.
I have that $\|T\|=\sup \{\|T f\|: f \in V \text { and }\|f\| \leq 1\}$ and $\|T \circ S \|=\sup \{\|T(Sf)\|: f \in V \text { and }\|f\| \leq 1\}$. But don't really know where to go from here. Thanks in advance 🙂
Best Answer
Hint: $$ \|T \circ S\| = \|S\| \cdot \sup\left\{\left\|T \left(\frac{Sx}{\|S\|} \right)\right\|: f \in V \text{ and } \|f\| = 1\right\}. $$