Page 101 - Textos de Matemática Vol. 44

ε
F (χ;ψ)=D ∥·∥
BIRKHOFF-JAMES ε-ORTHOGONALITY SETS 91
Proposition 4.1 and Theorem 4.2 yield readily the following. Corollary 4.3. Let χ,ψ ∈ X, with ψ ̸= 0. Then, for any ε ∈ (0,1),
Int Fε (χ;ψ) = μ∈C: ∥χ−λψ∥> 1−ε2∥ψ∥|μ−λ|, ∀λ∈C . ∥·∥
5. The case of norms induced by inner products
In the special case of norms induced by inner products, we can fully describe
the Birkhoff-James ε-orthogonality set F ε (χ; ψ). In particular, F ε (χ; ψ) is ∥·∥ ∥·∥
always a closed disk; this is the case for F 0.5 (χ; ψ) in the left part of Figure 1. ∥·∥2
Theorem 5.1 (For matrices, see [6, Section 5]). Let χ, ψ ∈ X with ψ ̸= 0 and ε ∈ [0, 1), and suppose that the norm ∥ · ∥ is induced by an inner product ⟨·, ·⟩. Then the Birkhoff-James ε-orthogonality set of χ with respect to ψ is the closed disk
⟨χ,ψ⟩ ⟨χ,ψ⟩ ε , χ− ψ √ .
∥ψ∥2 ∥ψ∥2 1 − ε2 ∥ψ∥ Proof. Ascalarμ∈CliesinFε (χ;ψ)ifandonlyif [4,7]
or equivalently, if and only if
∥·∥
ψ ⊥ε (χ − μψ),
|⟨ψ,χ−μψ⟩| ≤ ε∥ψ∥∥χ−μψ∥, or equivalently, if and only if
⟨ψ,χ−μψ⟩⟨χ−μψ,ψ⟩ ≤ ε2∥ψ∥2⟨χ−μψ,χ−μψ⟩, or equivalently, if and only if
|⟨χ,ψ⟩|2 ⟨ψ,χ⟩ ⟨χ,ψ⟩ 2 2 ∥χ∥2 ⟨ψ,χ⟩ ⟨χ,ψ⟩ 2 ∥ψ∥4 −μ ∥ψ∥2 −μ ∥ψ∥2 +|μ| ≤ ε ∥ψ∥2 −μ ∥ψ∥2 −μ ∥ψ∥2 +|μ| ,
or equivalently, if and only if
⟨χ,ψ⟩ 2
μ− (1−ε2) ≤ ∥ψ∥2
The proof is complete.
ε2 ⟨χ,ψ⟩ 2
χ−
∥ψ∥2 ∥ψ∥2
ψ .