Norm of Continuous Function is Continuous
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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.
$\blacksquare$