User:Anghel/Sandbox

Theorem
Let $\struct {X, \norm {\, \cdot \,}_X }$ be a normed vector space.

Let $\struct {X \times X, \norm {\, \cdot \,}_P }$ be the direct product of $X$ and $X$ with the direct product norm $\norm {\, \cdot \,}_P$.

Let $+ : X \times X \to X$ be the vector addition defined on $X$.

Then $+ : X \times X \to X$ is a continuous mapping.

Proof
Let $x_0, y_0 \in X$.

Let $\epsilon \in \R_{>0}$.

To show that $+$ is continuous, let $x, y \in X$ such that $\norm { x_0 - x }_X < \dfrac \epsilon 2$, and $\norm { y_0 - y }_X < \dfrac \epsilon 2$.

By definition of direct product norm, it follows that:


 * $\norm { \tuple{x_0,y_0} - \tuple{x,y} }_P = \map \max {\norm {x_0 - x}_X, \norm {y_0 - y}_X } < \dfrac \epsilon 2$