Theorem

Lax-Milgram Theorem

Analysis

The Lax-Milgram Theorem, named after Peter Lax and Arthur Milgram, who proved it in 1954, provides weak formulations for certain systems of equations posed on Hilbert spaces, the setting in which a solution is required to satisfy the equation only against a suitable space of test functions rather than in the strong, pointwise sense assumed by the original formulation. Weak formulations built this way carry many linear-algebra techniques over to the study of partial differential equations, and the Lax-Milgram Theorem supplies the essential existence and uniqueness guarantee that makes such weak solutions well defined.

Facts

Classification

Statement Form

Existence Theorem 1

Statement Form

Uniqueness Theorem 1

Statement Form

Identity or Equation 1

Statement

If a is a bounded and coercive bilinear form on a Hilbert space V, then for any bounded f in V', there is a unique solution u in V to the equation a(u,v) = f(v) for all v in V. 1

Proof Year

1954 1

Connections

Has Statement Form

In Branch

Functional Analysis

Sources

1. Lax-Milgram theorem (Wikipedia)

Lead section
The Lax-Milgram theorem, named after Peter Lax and Arthur Milgram who proved it in 1954, provides weak formulations for certain systems on Hilbert spaces.
General concept
named after Peter Lax and Arthur Milgram who proved it in 1954

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Lax-Milgram Theorem

Has Statement Form

In Branch