Theorems · Theorem · real analysis
range_deriv_subset_closure_span_image
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] (f : 𝕜 → F) {t : Set 𝕜},
Dense t → Set.range (deriv f) ⊆ closure ↑(Submodule.span 𝕜 (f '' t))Given a dense set t, then the range of deriv f is contained in the closure of the submodule
spanned by the image of t.
- Defined in
- Mathlib.Analysis.Calculus.Deriv.Slope
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- Set.imagestatement and proof · cited by 5,609
- Set.rangestatement and proof · cited by 4,705
- Set.univproof · cited by 3,945
- Submodule.spanstatement and proof · cited by 1,504
- closurestatement and proof · cited by 1,254
- derivstatement · cited by 676
Cited by1
Results whose statement or proof uses this declaration.
- stronglyMeasurable_deriv_with_paramproof · cited by 3