 Operator monotonicity

For a real Banach space $X$, a map $A:X\to X^{\ast }$ has operator monotonicity when $⟨A(u)-A(v),u-v⟩\ge 0$ for all $u,v\in X$. This is the monotone operator property, with the pairing between $X$ and its continuous dual space $X^{\ast }$. In a Hilbert space one can identify that pairing with the inner product.