The Lax-Milgram theorem in functional analysis is a representation theorem for bounded linear
functionals on a real Hilbert space 
. The result is used in the study of function
spaces and partial differential equations.
Let 
be a bounded coercive bilinear
form on a real Hilbert space 
. The Lax-Milgram theorem states that, for every bounded linear functional 
on 
, there exists a unique 
such that
 |
for all 
.
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 element 
necessarily corresponds to a unique element 
for which the inequality
 |
is satisfied for all elements 
where 
denotes the inner
product on 
.
