Suppose T∈L(V) is self-adjoint, λ∈F, and ϵ>0. Suppose there exists v∈V such that ∥v∥=1 and
∥Tv−λv∥<ϵ.
Prove that T has an eigenvalue λ′ such that ∣λ−λ′∣<ϵ.
This exercise shows that for a self-adjoint operator, a number that is close to satisfying an equation that would make it an eigenvalue is close to an eigenvalue.