In functional analysis, the Lax-Milgram theorem is a sort of representation theorem for bounded linear functionals on a Hilbert space . The result is of tantamount significance in the study of function spaces and partial differential equations.
Let be a bounded coercive bilinear form on a Hilbert space . The Lax-Milgram theorem states that, for every bounded linear functional on , there exists a unique such that
for all .
It is worth noting that the Lax-Milgram theorem follows immediately as a corollary of the Stampacchia theorem. One version of the Stampacchia theorem says that, under the assumptions above, any function necessarily corresponds to a unique function for which the inequality
is satisfied for all functions where here, denotes the inner product on .