If you have a norm defined on a vector space $V$ you can define the norm for the quotient by a subspace $W$:
|| v + W|| := \inf_{w \in W} || v + w ||
My question is: why does $W$ have to be a closed subspace?
In $\mathbb{R}^n$, as it happens, any subspace is closed, i.e. if something is a subspace then it's closed (e.g. lines or planes in $R^3$). Is this true for any vector space? I find it difficult to visualise subspaces of e.g. $L^1$.
Many thanks for your help!! (as always ; ) )