InfiniteSum

📁 Source: Mathlib/Analysis/Normed/Ring/InfiniteSum.lean

Statistics

Metric	Count
Definitions	0
Theorems mul_norm, mul_of_nonneg, hasSum_sum_range_mul_of_summable_norm, hasSum_sum_range_mul_of_summable_norm', summable_mul_of_summable_norm, summable_mul_of_summable_norm', summable_norm_sum_mul_antidiagonal_of_summable_norm, summable_norm_sum_mul_range_of_summable_norm, summable_of_absolute_convergence_real, summable_sum_mul_antidiagonal_of_summable_norm', summable_sum_mul_range_of_summable_norm', tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm, tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm', tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm, tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm', tsum_mul_tsum_of_summable_norm, tsum_mul_tsum_of_summable_norm'	17
Total	17

Summable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
mul_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	of_nonneg_of_le norm_nonneg norm_mul_le mul_of_nonneg
mul_of_nonneg 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SummationFilter.unconditional Pi.hasLe Real.instLE Pi.instZero Real.instZero	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Real.instMul SummationFilter.unconditional	—	summable_prod_of_nonneg mul_nonneg IsOrderedRing.toPosMulMono Real.instIsOrderedRing mul_left IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal tsum_mul_left TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace SummationFilter.instNeBotUnconditional mul_right

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
hasSum_sum_range_mul_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	HasSum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Finset.sum Finset.range Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum SummationFilter.unconditional	—	tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm Summable.hasSum Summable.of_norm summable_norm_sum_mul_range_of_summable_norm
hasSum_sum_range_mul_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	HasSum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Finset.sum Finset.range Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum SummationFilter.unconditional	—	tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm' Summable.hasSum summable_sum_mul_range_of_summable_norm'
summable_mul_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Summable ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	Summable.of_norm Summable.mul_norm
summable_mul_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	Summable ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	hasSum_of_subseq_of_summable Summable.mul_norm Filter.atTop_neBot SemilatticeSup.instIsDirectedOrder Filter.tendsto_finset_prod_atTop Filter.prod_atTop_atTop_eq Filter.Tendsto.prodMap Summable.hasSum Finset.sum_product Finset.sum_congr Filter.Tendsto.comp ContinuousAt.tendsto Continuous.continuousAt continuous_mul IsTopologicalSemiring.toContinuousMul IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing nhds_prod_eq HasSum.summable
summable_norm_sum_mul_antidiagonal_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Finset.HasAntidiagonal.antidiagonal Nat.instAddMonoid Finset.Nat.instHasAntidiagonal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	summable_sum_mul_antidiagonal_of_summable_mul T4Space.t3Space T5Space.toT4Space OrderTopology.t5Space instOrderTopologyReal IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal Summable.mul_of_nonneg norm_nonneg Summable.of_nonneg_of_le norm_sum_le Finset.sum_le_sum IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid norm_mul_le
summable_norm_sum_mul_range_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Finset.range Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	Finset.Nat.sum_antidiagonal_eq_sum_range_succ summable_norm_sum_mul_antidiagonal_of_summable_norm
summable_of_absolute_convergence_real 📖	mathematical	Real Filter.Tendsto Finset.sum Real.instAddCommMonoid Finset.range abs Real.lattice Real.instAddGroup Filter.atTop Nat.instPreorder nhds UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SummationFilter.unconditional	—	Summable.of_norm Real.instCompleteSpace hasSum_iff_tendsto_nat_of_nonneg norm_nonneg
summable_sum_mul_antidiagonal_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	Summable ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Finset.sum Finset.HasAntidiagonal.antidiagonal Nat.instAddMonoid Finset.Nat.instHasAntidiagonal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	summable_sum_mul_antidiagonal_of_summable_mul T4Space.t3Space instT4SpaceOfT1SpaceOfNormalSpace T2Space.t1Space TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace CompletelyNormalSpace.toNormalSpace UniformSpace.completelyNormalSpace_of_isCountablyGenerated_uniformity EMetric.instIsCountablyGeneratedUniformity IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing summable_mul_of_summable_norm'
summable_sum_mul_range_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	Summable ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing Finset.sum Finset.range Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing SummationFilter.unconditional	—	Finset.Nat.sum_antidiagonal_eq_sum_range_succ summable_sum_mul_antidiagonal_of_summable_norm'
tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing SummationFilter.unconditional Finset.sum Finset.HasAntidiagonal.antidiagonal Nat.instAddMonoid Finset.Nat.instHasAntidiagonal	—	Summable.tsum_mul_tsum_eq_tsum_sum_antidiagonal T4Space.t3Space instT4SpaceOfT1SpaceOfNormalSpace T2Space.t1Space TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace CompletelyNormalSpace.toNormalSpace UniformSpace.completelyNormalSpace_of_isCountablyGenerated_uniformity EMetric.instIsCountablyGeneratedUniformity IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing Summable.of_norm summable_mul_of_summable_norm
tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing SummationFilter.unconditional Finset.sum Finset.HasAntidiagonal.antidiagonal Nat.instAddMonoid Finset.Nat.instHasAntidiagonal	—	Summable.tsum_mul_tsum_eq_tsum_sum_antidiagonal T4Space.t3Space instT4SpaceOfT1SpaceOfNormalSpace T2Space.t1Space TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace CompletelyNormalSpace.toNormalSpace UniformSpace.completelyNormalSpace_of_isCountablyGenerated_uniformity EMetric.instIsCountablyGeneratedUniformity IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing summable_mul_of_summable_norm'
tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing SummationFilter.unconditional Finset.sum Finset.range	—	Finset.Nat.sum_antidiagonal_eq_sum_range_succ tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing SummationFilter.unconditional Finset.sum Finset.range	—	Finset.Nat.sum_antidiagonal_eq_sum_range_succ tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm'
tsum_mul_tsum_of_summable_norm 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing SummationFilter.unconditional	—	Summable.tsum_mul_tsum T4Space.t3Space instT4SpaceOfT1SpaceOfNormalSpace T2Space.t1Space TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace CompletelyNormalSpace.toNormalSpace UniformSpace.completelyNormalSpace_of_isCountablyGenerated_uniformity EMetric.instIsCountablyGeneratedUniformity IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing Summable.of_norm summable_mul_of_summable_norm
tsum_mul_tsum_of_summable_norm' 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedRing.toNorm SummationFilter.unconditional ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing tsum ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NormedRing.toNonUnitalNormedRing SummationFilter.unconditional	—	Summable.tsum_mul_tsum T4Space.t3Space instT4SpaceOfT1SpaceOfNormalSpace T2Space.t1Space TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace CompletelyNormalSpace.toNormalSpace UniformSpace.completelyNormalSpace_of_isCountablyGenerated_uniformity EMetric.instIsCountablyGeneratedUniformity IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing summable_mul_of_summable_norm'

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