Theorems · Theorem · approximation theory
Continuous.isBounded_range_iff_isBigO
∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] {D : Type u_2} [inst_1 : TopologicalSpace D] {f : D → E},
Continuous f → (Bornology.IsBounded (Set.range f) ↔ f =O[Filter.cocompact D] 1)A continuous function f has bounded range if and only if it is O(1) with respect to the
cocompact filter.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Set.ofPredproof · cited by 6,101
- Set.imageproof · cited by 5,609
- Norm.normproof · cited by 5,413
- Set.rangestatement and proof · cited by 4,705
- mul_oneproof · cited by 3,885
- Compl.complproof · cited by 2,925
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Continuousstatement and proof · cited by 2,592
- IsCompactproof · cited by 1,282
Cited by1
Results whose statement or proof uses this declaration.
- Continuous.isBounded_range_iff_isBigO_atTop_atBotproof · cited by 1