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A mapping $T$ on an inner-product space is firmly nonexpansive whenEvery firmly nonexpansive mapping is nonexpansive, and the resolvent of a maximal monotone operator is firmly nonexpansive.

The Hilbert space completion of an inner-product space $V$ is a Hilbert space containing an isometric dense copy of $V$. It can be constructed from Cauchy sequences in $V$, identifying two sequences when the norm of their difference tends to zero.

An operator $F$ on an inner-product space is monotone whenfor every $v,w$. The gradient of every differentiable convex function is monotone.

A real linear operator $\mathcal{L}$ on an inner-product space is a positive-definite operator when it is self-adjoint andfor every nonzero $u$ in its domain. In the variational setting one normally requires the stronger uniform estimate $⟨\mathcal{L}u,u⟩\ge c\Vert u{\Vert }_V^2$ for some $c>0$, which is coercivity.