Page 98 - Textos de Matemática Vol. 44

88 M. KARAMANLIS AND P. J. PSARRAKOS
Proof. Without loss of generality, we may assume that the region Ω is com-
pact. For the sake of contradiction, we also assume that for every ε ∈ [0, 1),
there is scalar με ∈ C such that με ∈/ F ε (χ; ψ). Then, there exist two sequences ∥·∥
{εn}n∈N ⊂ [0,1) and {μn}n∈N ⊂ Ω such that εn −→ 1− and μn ∈/ Fεn (χ;ψ) for ∥·∥
all n ∈ N. By the compactness of Ω, it follows that {μn}n∈N has a converging subsequence, say {μkn }n∈N ⊂ Ω, which converges to a μ ∈ Ω.
If μ ∈ F εˆ (χ; ψ) for some εˆ ∈ [0, 1), then by Theorem 3.1, and without loss ∥·∥
of generality, we may assume that μ lies in the interior of F εˆ (χ; ψ). Then there ∥·∥
is an n′ ∈ N such that μkn ∈ Fεˆ (χ;ψ) for every n ≥ n′. Moreover, there is ∥·∥
an n′′ ∈ N such that εkn > εˆ for every n ≥ n′′. As a consequence, for every n ≥ max{n′, n′′}, μkn ∈ F εˆ (χ; ψ) ⊆ F εkn (χ; ψ); this a a contradiction. So, for
∥·∥ ∥·∥
everyε∈[0,1),μ∈/Fε (χ;ψ).Thus,foreveryε = 1− 1 , n∈N,thereis ∥·∥ nn2
a scalar λn ∈ C such that
∥χ−(μ−λn)ψ∥ < 1− 1−n2 ∥ψ∥|λn| = n∥ψ∥|λn|,
1 2 1 |∥λnψ∥−∥χ−μψ∥| ≤ ∥λnψ−χ−μψ∥ < 1 ∥ψ∥|λn|,
or
(3.1) or
The bounded sequence λ2, λ3, . . . has a converging subsequence which converges to a scalar λ0 ∈ C. By (3.1), it follows
∥λknψ−χ−μψ∥ < 1 ∥ψ∥|λkn|, kn
n |λn|∥ψ∥ 1−n <∥χ−μψ∥.
1
Hence, for every n ≥ 2,
|λn| < ∥ψ∥ 1− 1 ≤ 2 ∥ψ∥ .
∥χ−μψ∥ ∥χ−μψ∥ n
and as n −→ +∞,
This is a contradiction because χ is not a scalar multiple of ψ.
{λkn }n∈N

∥λ0ψ−χ−μψ∥ = 0.