Quotient of a normed space

Remarks

The number $\norm{x + M}$ is the distance from $x$ to $M$. Closedness of $M$ is what makes this a norm: $\norm{x + M} = 0$ means that $x$ is in the closure of $M$, hence in $M$. The quotient map satisfies $\norm{\pi(x)} \le \norm{x}$ and maps the open unit ball of $X$ onto the open unit ball of $X/M$. If $X$ is a Banach space, then so is $X/M$. Example: in $\R^2$ with the Euclidean norm, let $M = \set{(a, 0) : a \in \R}$; then $\norm{(a, b) + M} = \inf_{c} \sqrt{(a + c)^2 + b^2} = \abs{b}$, which is a norm on $\R^2/M$. Example: in $\ell^\infty$ with $M = c_0$, one has $\norm{x + c_0} = \limsup_{n} \abs{x_n}$. Non-example: in $c_0$, the subspace $c_{00}$ of finitely supported sequences is dense but not closed; every $x \in c_0$ has $\norm{x + c_{00}} = 0$, so the formula gives only a seminorm on $c_0/c_{00}$.