How is this a bound: $|(\iota(Tx))(\varphi)| = |\varphi(Tx)| \leq \Vert \varphi \circ T\Vert \cdot \Vert x \Vert \leq \Vert \varphi \circ T \Vert$

Question

I'm wondering how this normed composition---from this answer---works as a bound. Please read question too.

$$|(\iota(Tx))(\varphi)| = |\varphi(Tx)| \leq \Vert \varphi \circ T\Vert \cdot \Vert x \Vert \leq \Vert \varphi \circ T \Vert \tag{A}$$

Specifically, the answer shows that $T$ is a bounded linear operator. But isn't it using the fact that $T$ is bounded in the normed composition occurring the end of (A)? If not, and this is indeed legitimate, would someone mind explaining in more detail how $\|\varphi T\|$ is a bound.

Answer 1

It is an assumption in that proof that $\phi \circ T$ is a bounded linear map, so it has a norm.