The above equalities are widely used in describing the operator norms. I am going over my (old) notes and I am stuck in why $\frac{Tx}{||x||}= T(\frac{x}{||x||})$. Could you please explain it to me? It seems that I used to get it 3 years ago, but now I am puzzled. I am sure it's under my nose, but appreciate why the 2nd part equals to the third part in the image above.
It's because\begin{align}T\left(\frac x{\lVert x\rVert}\right)&=T\left(\frac 1{\lVert x\rVert}x\right)\\&=\frac1{\lVert x\rVert}T(x)\\&=\frac{T(x)}{\lVert x\rVert}.\end{align}