I'm working on some problems related to norms of quotients spaces. There appear to be two definitions for the quotient space norm:
Let $Y$ be a closed subspace of a normed space of $(X,\|\cdot\|)$. Show that the norm $\|\cdot\|_0$ defined on $X/Y$ is given by $$\|\hat{x}\|_0 = \inf_{x\in\hat{x}}\|x\|$$ where $\hat{x} \in X/Y$.
And the one defined using the seminorm (properties referred to in the problem here)
Should I be able to prove that these norms are equivalent so that $p(x) = \inf_{x \in \hat{x}}\|x\|$?
This is related to this question.