Theorems · Theorem · general topology
ContinuousMap.norm_le
∀ {α : Type u_1} {E : Type u_3} [inst : TopologicalSpace α] [inst_1 : CompactSpace α]
[inst_2 : SeminormedAddCommGroup E] (f : C(α, E)) {C : ℝ}, 0 ≤ C → (‖f‖ ≤ C ↔ ∀ (x : α), ‖f x‖ ≤ C)The norm of a function is controlled by the supremum of the pointwise norms.
- Defined in
- Mathlib.Topology.ContinuousMap.Compact
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Norm.normstatement · cited by 5,413
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- ContinuousMapstatement and proof · cited by 2,491
- CompactSpacestatement and proof · cited by 593
- BoundedContinuousFunction.norm_leproof · cited by 11
Cited by13
Results whose statement or proof uses this declaration.
- IsGreatest.norm_cfcproof · cited by 6
- IsGreatest.norm_cfcₙproof · cited by 5
- isBigO_norm_Icc_restrict_atTopproof · cited by 2
- ContinuousMapZero.hasFiniteIntegral_of_boundproof · cited by 1
- isBigO_norm_Icc_restrict_atBotproof · cited by 1
- isBigO_norm_restrict_cocompactproof · cited by 1
- continuous_cfcHomSuperset_leftproof · cited by 1
- continuous_cfcₙHomSuperset_leftproof · cited by 1
- Real.integrable_of_summable_norm_Iccproof · cited by 1
- UnitAddTorus.mFourier_normproof · cited by 1
- ContinuousMap.hasFiniteIntegral_of_boundproof · cited by 1
- ContinuousMap.norm_eq_norm_coeFnproof · cited by 0