Invariant Subspace Problem
Show that every bounded linear operator T : H → H on a separable Hilbert space H of dimension at least 2 has a non-trivial closed T-invariant subspace: a closed linear subspace W of H, which is different from H and from {0}, such that `T ( W )...
Mathematical statement
Show that every bounded linear operator T : H → H on a separable Hilbert space H of dimension
at least 2 has a non-trivial closed T-invariant subspace: a closed linear subspace W of H,
which is different from H and from {0}, such that T ( W ) ⊂ W. One needs the assumption that
the dimension of H is at least 2 because otherwise any subspace would be either H or {0}.
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Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
Invariant_subspace_problem
theorem Invariant_subspace_problem [InnerProductSpace ℂ H] [TopologicalSpace.SeparableSpace H] [CompleteSpace H] (hdim : 2 ≤ Module.rank ℂ H) (T : H →L[ℂ] H) : Nonempty (ClosedInvariantSubspace T) := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References