Clarification of notation $\|fw\|$

Question

this is the question:

Show that for each linear map $f:\mathbb R^d → \mathbb R^e$ there exists $a < \infty$ so that $\|fw\|< a\|w\|$ for each $w$ in $\mathbb R^d.$

And my problem is that $f$ is a map so shouldn't it be $\|f(w)\|$ instead of $\|fw\|$? If not could you explain me how to understand this, please ?

Thank you.

Answer 1

When working with linear maps, one frequently omits brackets, for brevity. This shouldn't surprise you, after all we also write $\log x$ or $\sin x$ instead of $\log(x)$ and $\sin(x)$.

Answer 2

A linear map $f$ is continuous, and so is the function $y\mapsto\|y\|$. Therefore the real-valued function $\phi(x):=\|f(x)\|$ is bounded on the compact set $S^{d-1}\subset{\mathbb R}^d$ (the unit sphere): There is an $a>0$ such that $\bigl|\phi(x)\bigr|\leq a$ for all $x\in S^{d-1}$. From the linearity of $f$ it then follows that $\|f(x)\|\leq a\|x\|$ for all $x\in{\mathbb R}^d$.