Research proof index

The continuous spectrum, σᶜ, of a partial linear map. A complex number z is in σᶜ T iff the range of T - z • 1 is not closed. -/ def continuousSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ := {z : ℂ | ¬root.IsClosed ((T - z • 1).toFun.range : Set H)} @[inherit_doc continuousSpectrum] scoped notation "σᶜ" => continuousSpectrum lemma continuousSpectrum_eq (T...

If an operator is not closed then its resolvent set is empty.

The resolvent set is an open subset of ℂ.

Crucial inducion for formalized $\Sigma_1$-completeness.

!⟪x, y, z, ...⟫ notation for Seq -/ syntax "⟪" term,* "⟫" : term macro_rules | (⟪$term:term, $terms:term,*⟫) => (pair $term ⟪$terms,*⟫) | (⟪$term:term⟫) => ($term) @[app_unexpander pair] meta def pairUnexpander : Lean.PrettyPrinter.Unexpander | `($_ $term$term2) => `(⟪$term,$term2⟫) | _ => throw () lemma pair_graph {a b c : V} : c = ⟪a, b⟫ ↔ (a...

The determinant of the image of SL(2, ℂ) in the Lorentz group is one.