English
If a is an algebra over Real and σa is compact, then σℝ≥0a is compact; hence the NNReal spectrum is closed in its ambient topology.
Русский
Если a — алгебра над Real и σa компактно, то σℝ≥0a компактно; следовательно, спектр NNReal замкнут в своей топологии.
LaTeX
$$IsCompact(\\sigma_{\\mathbb{R}_{≥0}}(a))$$
Lean4
instance _root_.quasispectrum.instCompactSpaceNNReal [NormedSpace ℝ B] [IsScalarTower ℝ B B] [SMulCommClass ℝ B B]
(a : B) [CompactSpace (quasispectrum ℝ a)] : CompactSpace (quasispectrum ℝ≥0 a) :=
by
rw [← isCompact_iff_compactSpace] at *
rw [← quasispectrum.preimage_algebraMap ℝ]
exact isClosed_nonneg.isClosedEmbedding_subtypeVal.isCompact_preimage <| by assumption