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For the quadratic variational principle for a symmetric positive operator below, use a symmetric operator $L$ on the real Hilbert space $H$. Strict positive definiteness meansA uniformly positive definite symmetric operator satisfies the stronger coercive operator conditionIn this variational setting, “positive definite” is often used for a symmetric operator with this uniform bound. We will state explicitly where the coercive operator bound is needed. For a bounded linear operator defined on all of $H$, symmetry means that the operator is self-adjoint. For an unbounded operator, positivity is imposed on its operator domain, and the variational formulation is made on its form domain.

Symmetry is essential in a real Hilbert space: positivity of the quadratic expression alone does not imply symmetry. For example, $L=I+K$ with nonzero real skew-symmetric matrix $K$ satisfies $⟨Lv,v⟩=\Vert v{\Vert }^2$, but its quadratic functional has derivative involving $I$, not $I+K$. A positive definite symmetric operator supplies both the positivity and symmetry needed in part (b).

A symmetric operator $L$ on a real Hilbert space is strictly positive definite when $⟨Lv,v⟩>0$ for every nonzero $v$ in its operator domain. Strict positivity proves uniqueness of a solution of $Lu=f$, but in infinite dimension it need not prove existence for every $f$. A uniformly positive definite symmetric operator additionally has a positive lower bound independent of $v$. Symmetry must be stated separately when positivity is defined only by a real quadratic expression: adding a skew-symmetric matrix changes the operator without changing that expression.