Lions–Lax–Milgram theorem

A theorem ensuring that a bounded bilinear form defines a uniquely solvable variational problem when it satisfies coercivity or an appropriate inf-sup condition.

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Lax–Milgram theorem: A theorem guaranteeing a unique solution to a bounded coercive bilinear variational equation on a Hilbert space. The coercive case is the theorem’s classic existence-and-uniqueness result.
Hilbert space: A complete inner-product space, generalizing Euclidean geometry to possibly infinite dimensions. The classical theorem is stated for bounded forms on Hilbert spaces.
Weak solution: A solution satisfying a differential equation through an integral identity rather than pointwise derivatives. The variational equation often defines the weak solution whose existence the theorem guarantees.
Fredholm alternative: A theorem relating solvability of a linear operator equation to the kernels of the operator and its adjoint. It treats operator equations that may lack uniqueness, unlike coercive variational solvability.
Coercive bilinear form: A bilinear form whose value on a vector paired with itself bounds that vector’s squared norm from below. Coercivity supplies the direct solvability condition in the Hilbert-space setting.
Bounded bilinear form: A bilinear map whose magnitude is bounded by a constant times the norms of its two arguments. Boundedness controls the variational form and is a central theorem hypothesis.
Poisson equation: The elliptic partial differential equation relating the Laplacian of a function to a prescribed source. Its weak Dirichlet formulation is a standard coercive application.
Picard–Lindelöf theorem: A theorem guaranteeing local existence and uniqueness for ordinary differential equations with a Lipschitz vector field. It solves initial-value problems by iteration rather than coercivity of a bilinear form.
Inf-sup condition: A stability condition requiring a bilinear form’s action to remain uniformly detectable across paired function spaces. The inf-sup criterion extends solvability beyond coercive forms and supports mixed formulations.
Dual space: The space of continuous linear functionals on a given normed vector space. The variational equation represents its right-hand side as a continuous functional.