Consider $X$ a normed space with norm $\|\cdot\|$ and $M$ a closed subspace of $X$. In the quotient space $X/M$ we define the quotient norm:
$$|||\hat{x}||| = \inf_{y\in M} \|\hat{x}+y\|, \quad \hat{x}\in X/M.$$
I'm trying to prove the well-definiton of this norm, that is, given $\hat{x_1}$, $\hat{x_2}\in X/M$ such that $x_1-x_2\in M$ then it must ve verified that $|||\hat{x_1}|||=|||\hat{x_2}|||$.
By means of triangle inequality property of the norm $\|\cdot\|$ I managed to show that
$$|||\hat{x_1}|||-|||\hat{x_2}||| \leq \|x_1-x_2\|,$$
but that doesn't help to conclude what I want. I would really appreciate if someone can please help me with this.
Best Answer
Suppose $\hat{x}_1-\hat{x}_2=m\in M$. Since $M$ is a subspace, taking the $\inf$ over arbitrary $z\in M$ yields the same result as taking the $\inf$ over $m-y$ for arbitrary $y\in M$ (since $m-y$ is still in $M$ and any value at $z$ is achieved when $y$ is $m-z$). Using this change of variables ($z$ for $m-y$),
$$|||\hat{x}_1|||=\inf_{z\in M}\|\hat{x}_1-z\|=\inf_{y\in M}\|\hat{x}_1-m+y\|=\inf_{y\in M}\|\hat{x}_2+y\|=|||\hat{x}_2|||$$