Theorem. Let $(X,\| \cdot \|$) be a normed space. Then $$ p(x+U) = \inf_{z \in U} \|z-x \|$$ defines a semi-norm on $X/U$ with $p(x+U) \leq \|x \|$.
a) If $U$ is closed, then $p$ is a norm.
b) If $U$ is closed and $X$ a Banach space, then $(X/U, p)$ is a Banach space.
I'd like some intuition on a), b). (Not a proof!) What is a deeper reason that closedness fixes the semi-norm and that Banach spaces induce Banach spaces?
I suppose amongst the most important applications is to take $U$ to be some kernel (even of a semi-norm, as seen in $L^p$ spaces). Perhaps it's easier to argue intuitively here and then to take some natural generalization?
(1) $p$ is always a seminorm on the quotient and if it is not a norm there is $y=q(x)$ (where $q$ is the quotient map) such that $p(y)=0$ but $y\neq 0$. The latter condition means that $x$ isn't in $U$ but the former means that one can approximate $x$ arbitrarily well by elements of $U$.
(2) Completeness of quotients is much deeper than (1). One can deduce it from a version of the open mapping theorem (a linear continuous map between Banach space is surjective and open if the closure of the image of the unit ball of the domain contains some ball in the range, see, e.g., Rudin's book on functional analysis) applied to $j\circ q$ where $j$ is the embedding of $X/U$ into its completion.