Motivation for the norm of the quotient space

I'm studying Functional Analysis, more specifically, the quotient space. Since more often we see the $\sup$ as some norm definitions, I was wondering, what is the motivation for the quotient norm to be defined as the $\inf$.

Given $X$ a Bananch space and $M$ a closed subspace of $X$, we define a norm $\parallel \ \parallel_{Q}$ on the quotient space $Q = X/M$ by: $$\parallel [x]\parallel_{Q} = \inf_{y\in [x]}\parallel y \parallel = \inf_{m\in M}\parallel x+m \ \parallel $$

Thanks for the help.

Solution 1:

You want the quotient space $X/Y$ to have the following universal property: There exists a contraction $\pi\colon X\to X/Y$, and if $T\colon X\to Z$ is a contraction with $Y\subset \ker T$, then there exists a unique contraction $\tilde T\colon X/Y\to Z$ such that $\tilde T\circ\pi=T$. Of course the map $\tilde T$ is just given by $\tilde T(x+Y)=Tx$.

This is true in the category of vector spaces (if you forget about norms and contractivity of the maps), and the quotient norm is the unique norm that makes it true in the category of Banach spaces and contractive linear maps.