Is Closable of exists dense formal Adjoint
LinearPMap.isClosable_of_exists_dense_formalAdjoint
Plain-language statement
A LinearPMap with densely-defined formal adjoint is closable.
Exact Lean statement
lemma isClosable_of_exists_dense_formalAdjoint [CompleteSpace H] [CompleteSpace H']
(h : U.HasDenseDomain) (h_fadj : ∃ U' : H' →ₗ.[ℂ] H, U'.HasDenseDomain ∧ U'.IsFormalAdjoint U) :
U.IsClosableFormal artifact
Lean source
lemma isClosable_of_exists_dense_formalAdjoint [CompleteSpace H] [CompleteSpace H'] (h : U.HasDenseDomain) (h_fadj : ∃ U' : H' →ₗ.[ℂ] H, U'.HasDenseDomain ∧ U'.IsFormalAdjoint U) : U.IsClosable := by have h_adj : U†.HasDenseDomain := by obtain ⟨U', hU', hU''⟩ := h_fadj exact hU'.mono (hU''.symm.le_adjoint h).1 use U†† ext rw [adjoint_graph_eq_graph_adjoint h_adj, adjoint_graph_eq_graph_adjoint h, mem_submodule_adjoint_adjoint_iff_mem_submoduleToLp_orthogonal_orthogonal, orthogonal_orthogonal_eq_closure, mem_submodule_iff_mem_submoduleToLp, submoduleToLp_closure]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/Unbounded.lean:234-244
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