Norms are Continuous

Last updated Nov 6, 2022

Let $(X, ||\cdot||)$ be a Normed Vector Space. Then $||\cdot||$ is a Continuous Function.

Recall $x,y \mapsto ||x - y||$ is the distance function induced by our norm. Let $x \in X$. Let $x_{n} \to x$ be a convergent sequence in $X$. Then $(x_{n}, 0) \to (x, 0)$ in $X \times X$ because Components Converge iff Sequence Converges in Finite Product Metric Spaces. Because Metrics are Continuous and Continuous Functions Preserve Sequence Limits, $||x_{n}|| = ||x_{n} - 0|| \to ||x - 0|| = ||x||$. By the definition of Continuous Function in a Metric Space, $||\cdot||$ is a Continuous Function. $\blacksquare$