= Solution
The <quotient norm> is
$$
\boxed{\|x+Y\|_{X/Y}=\inf_{y\in Y}\|x-y\|.}
$$
Changing the representative does not change the infimum. The <triangle inequality> and absolute homogeneity follow from those of the original <norm>. Vanishing means $x$ belongs to the closure of $Y$, hence to $Y$ because the subspace is closed. Thus this is a <norm>.
Use the absolutely convergent series criterion for completeness. If $\sum\|\xi_n\|_{X/Y}<\infty$, choose representatives $z_n$ of $\xi_n$ satisfying $\|z_n\|\leq\|\xi_n\|_{X/Y}+2^{-n}$. The series $\sum z_n$ converges in the <Banach space> $X$ to some $z$. Since the <quotient map> is contractive, $\sum\xi_n$ converges to $z+Y$. Consequently \b[the quotient is a <Banach space>].
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