Page 18 - Textos de Matemática Vol. 34

8 DAVID E. EDMUNDS
Granted this for the moment, it follows that
k ≤ μ0( γp (b − a), I) ≤ ν0( γp (b − a), I) ≤ k,
2k 2k
and so ak(E0) = γp (b − a), as required. To justify our claim, let S be a subspace
2k 00
of W1p(I) of finite dimension d(S) and let P : W1p(I) → Lp(I) be a bounded linear operator of rank r(P ) < d(S). Let {e1, ..., ed(S)} be a basis of S. Then there exist λ1, ..., λd(S), not all zero, such that
Put ψ = d(S) λjej. Then j=1
and so
Hence
∥(E0 −P)ψ|Lp(I)∥=∥E0ψ|Lp(I)∥≥θ(S)−1∥ψ′ |Lp(I)∥, 01 −1
E0−P|Wp(I)→Lp(I)≥θ(S) .
P
λjej
= 0.
d(S) j=1

ad(S)(E0) ≥ θ(S)−1,
which shows that aμ0(ε,I)(E0) ≥ ε and establishes the claim.
The proof of Theorem 2.2 and of the results leading up to it follows that given in [11]; the result was independently proved in [9]. For a connected account of the whole matter see [7].
Remark 2.2. Similar procedures (see [7], [9] and [11]) show that the approximation numbers of the embedding EC : W1 (I) → Lp,C(I) are given by the same formula:
p,C
ak(EC) = γp(b − a) (k ∈ N).
2k
When Ω is a bounded domain in Rn and p = 2, it is well known that the 0
approximation numbers of the embedding E0(Ω) : W12(Ω) → L2(Ω) are given by (2.11) ak(E0) = λ−1/2 (k ∈ N),
k
where λk is the kth eigenvalue of the Dirichlet Laplacian on Ω. Since the asymptotic behaviour of the λk is known, it follows that limk→∞ k1/nak(E0) exists. Moreover, since when n = 1 the Dirichlet eigenvalues are known precisely, this relationship between approximation numbers and eigenvalues gives exact fomulae for the ap- proximation numbers, just as in Theorem 2.2. It turns out (see [9]) that when