Is Symmetric closure
LinearPMap.IsSymmetric.closure
Plain-language statement
The closure of a symmetric densely-defined operator is symmetric: T†† is a symmetric closed extension of T, so it extends T.closure, whose symmetry then descends.
Exact Lean statement
lemma IsSymmetric.closure [CompleteSpace H] (hsym : T.IsSymmetric) (hdense : T.HasDenseDomain) :
T.closure.IsSymmetricFormal artifact
Lean source
lemma IsSymmetric.closure [CompleteSpace H] (hsym : T.IsSymmetric) (hdense : T.HasDenseDomain) : T.closure.IsSymmetric := by have hle : T ≤ T† := (isSymmetric_def.mp hsym).le_adjoint hdense have hadj_dense : T†.HasDenseDomain := hdense.mono hle.1 have hT_le : T ≤ T†† := (adjoint_isFormalAdjoint hdense).le_adjoint hadj_dense have hc : (T††).IsClosed := adjoint_isClosed hadj_dense have h1 : (T††).IsSymmetric := (isSymmetric_iff_le_adjoint (hdense.mono hT_le.1)).mpr (adjoint_antitone (Or.inl (hdense.mono hT_le.1)) (adjoint_antitone (Or.inl hdense) hle)) exact h1.of_le (hc.closure_eq ▸ hc.isClosable.closure_mono hT_le)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/Unbounded.lean:854-863
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