confirmation of an equality regarding norm of a vector-valued function

Question

Let $|\cdot|$ denote absolute value of a real number. For an $R^2-$ valued function $f=\begin{bmatrix}f_{1}(x),f_{2}(x)\end{bmatrix}^{\top}$, define its norm as

$||f||=max\{\underset{x}{sup}|f_{1}(x)|,\underset{x}{sup}|f_{2}(x)|\}$.

Then it's true that

$||f||^2=(max\{\underset{x}{sup}|f_{1}(x)|,\underset{x}{sup}|f_{2}(x)|\})^2=max\{\underset{x}{sup}|f_{1}(x)|^2,\underset{x}{sup}|f_{2}(x)|^2\}$, right?

Answer 1

• If $a,b$ are nonnegative, then $(\max\{a,b\})^2 = \max\{a^2, b^2\}$
• If $g$ is nonnegative, then $(\sup_x \{g(x)\})^2 = \sup_x \{g(x)\}^2$