It is understood that this equation must hold for all test functions,
, which must vanish at the boundaries where
. The Galerkin approximation 
to the weak form
solution 
in (17) can be expressed as:
The trial functions 
form a basis for an N+1
dimensional space 
. We define the Galerkin approximation to
be that element 
which satisfies:
Since our differential operator A is positive definite and self
adjoint (i.e., 
for some non-zero
positive constant 
and
, respectively), then we
can define a space E with an inner product defined as
and norm (the so-called energy norm) equal to:
