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 ; ) )
Best Answer
Suppose $v$ is in the closure of $W$. Then there exists a sequence $w_n$ of points in $W$ such that $||v-w_n||$ tends to 0. Since $W$ is a subspace, $-w_n$ is in $W$, so $||v+W||=\inf_{w \in W} ||v+w||=0$. If we want our norm on the quotient space to actually be a norm, rather than just a seminorm, then we need $||v +W|| =0$ to imply $v+W=0$, i.e. $v \in W$. So we need everything in the closure of $W$ to be in $W$, i.e. we need $W$ to be closed.
There certainly are natural examples of non-closed subspaces of normed spaces: For example, consider $L^1([0,1])$. The continuous functions form a proper subspace which is dense, and so not closed.