Theorems · Theorem · functional analysis
smul_closedBall
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NormedField 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E]
(c : 𝕜) (x : E) {r : ℝ}, 0 ≤ r → c • Metric.closedBall x r = Metric.closedBall (c • x) (‖c‖ * r)- Cited by
- 6 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Norm.normstatement · cited by 5,413
- MulZeroClass.zero_mulproof · cited by 1,625
- eq_or_neproof · cited by 1,117
- NormedFieldstatement and proof · cited by 1,084
- zero_smulproof · cited by 716
- Metric.closedBallstatement and proof · cited by 704
- Set.smulSetstatement · cited by 608
- norm_zeroproof · cited by 366
Cited by6
Results whose statement or proof uses this declaration.
- MeasureTheory.addHaar_image_le_mul_of_det_ltproof · cited by 4
- smul_unitClosedBallproof · cited by 3
- MeasureTheory.Measure.addHaar_closedBall_mulproof · cited by 1
- tendsto_integral_comp_smul_smul_of_integrableproof · cited by 1