Quotient of a normed space

Summary

The quotient of a normed space $X$ by a closed subspace $M$ is the quotient vector space $X/M$. Its norm measures the distance of a vector to $M$. It is a normed space, and it is a Banach space when $X$ is. Quotients appear whenever one ignores a subspace. Examples are the open mapping theorem, the study of the kernel of an operator, and duality, where $(X/M)^*$ is identified with the annihilator of $M$.