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A coercive form need not be symmetric

Statement refuted

Assume Countable Choice for the Lax--Milgram conclusion. On $H = C^{2}$ with the standard inner product define $a (u, v) : = u_{1} \overline{v_{1}} + u_{2} \overline{v_{2}} + i u_{1} \overline{v_{2}} .$ Then $a$ is sesquilinear in the convention of Bounded, coercive and symmetric sesquilinear forms, bounded with $∣ a (u, v) ∣ \leq 2 ∥ u ∥ ∥ v ∥$ and coercive with $Re a (u, u) \geq \frac{1}{2} ∥ u ∥^{2}$ , so $α = \frac{1}{2}$ . It is not symmetric: with $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ one has $a (e_{1}, e_{2}) = i$ while $\overline{a (e_{2}, e_{1})} = 0$ . Hence the Lax--Milgram theorem The Lax--Milgram theorem applies to this nonsymmetric form, and symmetry is not needed for existence and uniqueness; the energy-minimisation corollary Symmetric Lax--Milgram is energy minimisation is the part that genuinely uses symmetry. The example is consistent with the abstract forcing remark Nonsymmetric Lax--Milgram is not a scalar minimisation principle.

Facts & Assumptions

Given: Countable Choice; the Hilbert space $H = C^{2}$ with the standard inner product $∥ u ∥^{2} = ∣ u_{1} ∣^{2} + ∣ u_{2} ∣^{2}$ ; the form $a (u, v) = u_{1} \overline{v_{1}} + u_{2} \overline{v_{2}} + i u_{1} \overline{v_{2}}$ ; and the vectors $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ .

[F1]

Sesquilinearity in the convention linear in the first argument and conjugate-linear in the second, with boundedness and coercivity as in Bounded, coercive and symmetric sesquilinear forms (Real and complex inner-product spaces and their induced length, Hilbert space).

[F2]

Scalar facts: $∣ z \overline{w} ∣ = ∣ z ∣ ∣ w ∣$ , $\overline{i} = - i$ , $i^{2} = - 1$ ; and for vectors in $C^{2}$ , $∣ z_{1} w_{1} ∣ + ∣ z_{2} w_{2} ∣ \leq (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2})^{1 / 2} (∣ w_{1} ∣^{2} + ∣ w_{2} ∣^{2})^{1 / 2}$ by Cauchy--Schwarz (Real and imaginary parts, complex conjugation, and modulus, Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F3]

Lax--Milgram applies to bounded coercive forms and does not assume symmetry; the energy-minimisation corollary does assume it (The Lax--Milgram theorem, Symmetric Lax--Milgram is energy minimisation, Nonsymmetric Lax--Milgram is not a scalar minimisation principle).

Proof

1.1F1F2

Sesquilinearity, boundedness and coercivity: for scalars $λ$ , $a (λ u, v) = λ a (u, v)$ and $a (u, λ v) = \overline{λ} a (u, v)$ directly from the definition, so $a$ is sesquilinear in the stated convention. Moreover $∣ a (u, v) ∣ \leq ∣ u_{1} ∣ ∣ v_{1} ∣ + ∣ u_{2} ∣ ∣ v_{2} ∣ + ∣ u_{1} ∣ ∣ v_{2} ∣$ , and Cauchy--Schwarz applied to the pairs $(∣ u_{1} ∣, ∣ u_{2} ∣)$ , $(∣ v_{1} ∣, ∣ v_{2} ∣)$ gives $∣ a (u, v) ∣ \leq (∣ u_{1} ∣ + ∣ u_{2} ∣) (∣ v_{1} ∣ + ∣ v_{2} ∣) \leq 2 ∥ u ∥ ∥ v ∥$ ; and $a (u, u) = ∣ u_{1} ∣^{2} + ∣ u_{2} ∣^{2} + i u_{1} \overline{u_{2}}$ has real part at least $∥ u ∥^{2} - ∣ u_{1} ∣ ∣ u_{2} ∣ \geq \frac{1}{2} ∥ u ∥^{2}$ , using $2 ∣ u_{1} ∣ ∣ u_{2} ∣ \leq ∣ u_{1} ∣^{2} + ∣ u_{2} ∣^{2}$ . Thus $a$ is coercive with constant $α = \frac{1}{2}$ .

1.2F1F2

Nonsymmetry: for the standard basis vectors, $a (e_{1}, e_{2}) = i$ while $\overline{a (e_{2}, e_{1})} = \overline{0} = 0$ , so $a (e_{1}, e_{2}) \neq \overline{a (e_{2}, e_{1})}$ and $a$ is not symmetric.

2.1F1F3step 1.1step 1.2∎

Consequences: $a$ is bounded and coercive but not symmetric, so Lax--Milgram applies and gives existence and uniqueness of solutions for every bounded conjugate-linear datum, while the energy-minimisation characterisation, which requires symmetry, does not apply. Thus symmetry is not needed for solvability but is genuinely used by the variational principle.

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Sources

Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer Universitext, 2011, complete 614-page text) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)