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 
.
