The solution of (15) satisfies:

and the approximate Galerkin solution obtained by solving (19) satisfies:

Subtracting (21) from (22) yields:

The difference denotes the error between the solution in the
infinite dimensional space **V** and the **N+1** dimensional space .
Equation (23) states that the error is orthogonal to all basis
functions spanning the space of possible Galerkin solutions. Consequently,
the error is orthogonal to all elements in and must therefore be
the minimum error. Thus the Galerkin approximation is an orthogonal
projection of the true solution onto the given finite dimensional
space of possible approximate solutions. Therefore, the Galerkin
approximation is the best approximation in the energy space **E**.