Theorems · Theorem · functional analysis
tendsto_iff_norm_sub_tendsto_zero
∀ {α : Type u_1} {E : Type u_4} [inst : SeminormedAddCommGroup E] {f : α → E} {a : Filter α} {b : E},
Filter.Tendsto f a (nhds b) ↔ Filter.Tendsto (fun e => ‖f e - b‖) a (nhds 0)- Defined in
- Mathlib.Analysis.Normed.Group.Continuity
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddCommGroup
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.
- Realstatement · cited by 25,697
- Filterstatement and proof · cited by 8,121
- nhdsstatement and proof · cited by 5,554
- Norm.normstatement · cited by 5,413
- Filter.Tendstostatement and proof · cited by 3,814
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- dist_eq_norm_subproof · cited by 29
- tendsto_iff_dist_tendsto_zeroproof · cited by 16
Cited by17
Results whose statement or proof uses this declaration.
- MeasureTheory.tendsto_setToFun_of_dominated_convergenceproof · cited by 5
- NormedRing.inverse_continuousAtproof · cited by 3
- ContinuousLinearMap.exists_preimage_norm_leproof · cited by 2
- tendsto_setIntegral_peak_smul_of_integrableOn_of_tendstoproof · cited by 2
- VitaliFamily.ae_tendsto_averageproof · cited by 2
- integrableOn_peak_smul_of_integrableOn_of_tendstoproof · cited by 2
- MeasureTheory.tendsto_lintegral_norm_of_dominated_convergenceproof · cited by 2
- RKHS.tendstoUniformlyOn_of_norm_kerFun_leproof · cited by 1
- Real.hasSum_pow_div_log_of_abs_lt_oneproof · cited by 1
- MeasureTheory.continuous_integral_integralproof · cited by 1
- ProbabilityTheory.strong_law_ae_of_measurableproof · cited by 1
- ProbabilityTheory.Kernel.continuous_integral_integralproof · cited by 1