Quotient norm ($\Vert x+Y\Vert ={\inf }_{y\in Y}\Vert x+y\Vert$)

The quotient norm measures the smallest norm among representatives of a coset. Its infimum need not be attained. Nevertheless, for every $\epsilon >0$, an element $z\in X/Y$ has a representative $v\in X$ with $\Vert v\Vert <\Vert z\Vert +\epsilon$. The quotient map is contractive.