The equation
y
=
f
+
K
y
y = f + Ky
is considered in a separable Hilbert space H, with K assumed compact and linear. It is shown that every approximation to y of the form
y
1
n
=
Σ
n
a
n
i
u
i
{y_{1n}} = {\Sigma ^n}{a_{ni}}{u_i}
(where {
u
i
{u_i}
} is a given complete set in H, and the
a
n
i
,
1
⩽
i
⩽
n
{a_{ni}},1 \leqslant i \leqslant n
, are arbitrary numbers) is less accurate than the best approximation of the form
y
2
n
=
f
+
Σ
n
b
n
i
K
u
i
{y_{2n}} = f + {\Sigma ^n}{b_{ni}}K{u_i}
, if n is sufficiently large. Specifically it is shown that if
y
1
n
{y_{1n}}
is chosen optimally (i.e. if the coefficients
a
n
i
{a_{ni}}
are chosen to minimize
‖
y
−
y
1
n
‖
\left \| {y - {y_{1n}}} \right \|
), and if
y
2
n
{y_{2n}}
is chosen to be the first iterate of
y
1
n
{y_{1n}}
, i.e.
y
2
n
=
f
+
K
y
1
n
{y_{2n}} = f + K{y_{1n}}
, then
‖
y
−
y
2
n
‖
⩽
α
n
‖
y
−
y
1
n
‖
\left \| {y - {y_{2n}}} \right \| \leqslant {\alpha _n}\left \| {y - {y_{1n}}} \right \|
, with
α
n
→
0
{\alpha _n} \to 0
. A similar result is also obtained, provided the homogeneous equation
x
=
K
x
x = Kx
has no nontrivial solution, if instead
y
1
n
{y_{1n}}
is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms
y
3
n
,
y
4
n
,
…
{y_{3n}},{y_{4n}}, \ldots
obtained by further iteration is also shown to be valid, if the range of K is dense in H.