= Solution
A <quotient operator> is a <surjective> <bounded linear operator> that is an <open map>. For a <linear map>, being an <open map> is equivalent to the existence of $c>0$ such that
$$
B_Y(0,c)\subseteq T(B_X(0,1)).
$$
The inclusion gives a neighbourhood of $Tx$ inside the <image> of every sufficiently small ball about $x$, by translation and scaling. Conversely, an <open map> sends $B_X(0,1)$ to a neighbourhood of zero. A linear <open map> onto $Y$ is automatically <surjective>, since its <image> contains a ball and is a <linear subspace>.
The <Banach-space adjoint> is $T^*g=g\circ T$. For $g\in Y^*$, the ball inclusion gives
$$
\|T^*g\|=\sup_{\|x\|<1}|g(Tx)|\geq\sup_{\|y\|<c}|g(y)|=c\|g\|.
$$
Also $\|T^*g\|\leq\|T\|\|g\|$. Thus \b[the <Banach-space adjoint> is an <isomorphic embedding>], with lower bound $c$.
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