Quotient Norm

Last updated Nov 1, 2022

Let $(X, ||\cdot||)$ be a Normed Vector Space and let $Y \subset X$ be a Closed Vector Subspace. Then $||[\cdot]|| : X / Y \to \mathbb{R}{\geq 0}$ defined as $$||[x]|| := \inf\limits{y \in Y} ||x - y|| = \text{dist}{x, Y}$$ for $x \in X$ is a Norm on $X / Y$.

TODO

1. The requirement that $Y$ is Closed is very important. See StackExchange on why the subspace needs to be closed.

• Real Numbers
• Set Distance