= Solution
Using the <dual pairing>, <monotone operator>[monotonicity] means
$$
\boxed{\langle A(u)-A(v),u-v\rangle\ge0\qquad(u,v\in X).}
$$
<strictly monotone operator>[Strict monotonicity] requires strict positivity whenever $u\ne v$. Coercivity with respect to zero, in the growth sense used for monotone-operator existence theory, means
$$
\boxed{\frac{\langle A(v),v\rangle}{\|v\|_X}\longrightarrow+\infty\qquad\text{as }\|v\|_X\to\infty.}
$$
This is a condition on all <sequences> escaping to <infinity>, not just each fixed direction with a bound depending on that direction. <strictly monotone operator>[Strict monotonicity] concerns pairs of distinct arguments; it does not by itself imply this uniform large-norm growth.
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