English
The same continuity property as in 42919 holds for the continuous character space to spectrum map.
Русский
Те же свойство непрерывности сохраняется для отображения характер-пр спектр.
LaTeX
$$continuous_characterSpaceToSpectrum (x) : Continuous (characterSpaceToSpectrum x)$$
Lean4
theorem nnreal_add {a b : A} (ha₁ : IsSelfAdjoint a) (hb₁ : IsSelfAdjoint b)
(ha₂ : SpectrumRestricts a ContinuousMap.realToNNReal) (hb₂ : SpectrumRestricts b ContinuousMap.realToNNReal) :
SpectrumRestricts (a + b) ContinuousMap.realToNNReal :=
by
rw [SpectrumRestricts.nnreal_iff_nnnorm (ha₁.add hb₁) (nnnorm_add_le a b), NNReal.coe_add, map_add, add_sub_add_comm]
refine nnnorm_add_le _ _ |>.trans ?_
gcongr
all_goals
rw [← SpectrumRestricts.nnreal_iff_nnnorm] <;>
first
| rfl
| assumption