Lax–Milgram is owned by the PDE track

The seam contract of this track assigns the Lax–Milgram theorem and the vocabulary of bounded and coercive sesquilinear forms to the partial-differential-equations development, and this page does not state them. What this page supplies is Riesz representation for Hilbert spaces, which the PDE track may cite, together with the bounded-operator vocabulary of A bounded linear operator between normed spaces: on a real or complex Hilbert space every bounded linear functional is represented by a vector, and that is the ingredient from which a later page proves Lax–Milgram after defining its own forms, boundedness, coercivity and continuity hypotheses.

Nothing on the present page assumes coercivity, symmetry or continuity of a sesquilinear form, and no representation of a form by an operator is asserted. Assuming the Axiom of Countable Choice, the Riesz theorem available here carries that hypothesis, and any later consumer that invokes it inherits the same assumption (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).