Norm of Continuous Function is Continuous

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.

Let $f : X \to Y$ be a continuous mapping.

Define $\norm f_Y : X \to \hointr 0 \infty$ by:

$\map {\paren {\norm f_Y} } x = \norm {\map f x}_Y$

for each $x \in X$.

Then $\norm f_Y$ is continuous.

Proof

Follows immediately from combining Norm on Vector Space is Continuous Function and Composite of Continuous Mappings is Continuous.

$\blacksquare$