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Ordinary Differential Equations 199
Theorem 9.4.1 - Let X be a Hilbert space. Let B : X × X → R be a bilinear form
and F : X → R be a linear operator, both satisfying the hypotheses of theorem 9.3.1.
Let us denote by U ∈ X the unique solution of the problem B(U,φ) = F(φ), ∀φ ∈ X.
For any n ∈ N, let Xn ⊂ X be a finite dimensional subspace of dimension n and
Fn : Xn → R be a linear operator. Let us denote by pn ∈ Xn the solution of the
problem B(pn,φ) = Fn(φ), ∀φ ∈ Xn.
Then, we can find a constant C > 0 such that, for any n ∈ N, we have
(9.4.4) U − pnX ≤ C
inf
χ∈Xn
U − χX + sup
φ∈Xn
φ≡0
Proof - For any n ∈ N, we define χn ∈ Xn such that
(9.4.5) B(χn,φ) = B(U,φ), ∀φ ∈ Xn.
|F(φ) − Fn(φ)|
.
φX
Existence and uniqueness of χn are obtained by applying theorem 9.3.1, after noting
that B(U, ·) : X → R is a linear operator for any fixed U ∈ X. We use the triangle
inequality to write
(9.4.6) U − pnX ≤ U − χnX + χn − pnX, n ∈ N.
Due to (9.3.6) and (9.4.5), for any χ ∈ Xn, we get
(9.4.7) U − χn 2 X ≤ C −1
2 B(U − χn,U − χn)
= C −1
2 B(U − χn,U − χ) ≤ C1C −1
2 U − χnX U − χX,
which gives an estimate of the error U − χn, n ∈ N.
On the other hand, (9.3.6) yields
(9.4.8) χn − pnX ≤ B(χn − pn,χn − pn)
C2 χn − pnX
≤