Documentation Verification Report

WithSeminorms

πŸ“ Source: Mathlib/Analysis/LocallyConvex/WithSeminorms.lean

Statistics

MetricCount
DefinitionsPolynormableSpace, SeminormFamily, addGroupFilterBasis, basisSets, comp, moduleFilterBasis, sigma, WithSeminorms
8
TheoremswithSeminorms_induced, toLocallyConvexSpace, toLocallyConvexSpace', iInf, induced, inf, sInf, withSeminorms, withSeminorms', bound_of_continuous, bound_of_continuous_normedSpace, const_isBounded, cont_normedSpace_to_withSeminorms, cont_withSeminorms_normedSpace, continuous_from_bounded, continuous_iff_continuous_comp, continuous_of_continuous_comp, isBounded_const, isBounded_sup, map_eq_zero_of_norm_eq_zero, map_eq_zero_of_norm_zero, basisSets_add, basisSets_iff, basisSets_intersect, basisSets_mem, basisSets_mem_nhds, basisSets_neg, basisSets_nonempty, basisSets_singleton_mem, basisSets_smul, basisSets_smul_left, basisSets_smul_right, basisSets_univ_mem, basisSets_zero, comp_apply, filter_eq_iInf, finset_sup_comp, withSeminorms_iff_nhds_eq_iInf, withSeminorms_iff_topologicalSpace_eq_iInf, withSeminorms_iff_uniformSpace_eq_iInf, withSeminorms_of_hasBasis, withSeminorms_of_nhds, polynormableSpace, withSeminorms, T1_of_separating, congr_equiv, continuousSMul, continuous_seminorm, equicontinuous_TFAE, finset_sups, firstCountableTopology, hasBasis, hasBasis_ball, hasBasis_zero_ball, image_isVonNBounded_iff_finset_seminorm_bounded, image_isVonNBounded_iff_seminorm_bounded, isOpen_iff_mem_balls, isVonNBounded_iff_finset_seminorm_bounded, isVonNBounded_iff_seminorm_bddAbove, isVonNBounded_iff_seminorm_bounded, mem_nhds_iff, partial_sups, separating_iff_T1, separating_of_T1, tendsto_nhds, tendsto_nhds', tendsto_nhds_atTop, toLocallyConvexSpace, toPolynormableSpace, topologicalAddGroup, topology_eq_withSeminorms, uniformEquicontinuous_iff_bddAbove_and_continuous_iSup, uniformEquicontinuous_iff_exists_continuous_seminorm, withSeminorms_eq, instLocallyConvexSpaceRealOfPolynormableSpace, instPolynormableSpace, instPolynormableSpaceForall, instPolynormableSpaceProd, instPolynormableSpaceSubtypeMemSubmodule, norm_withSeminorms, withSeminorms_iInf, withSeminorms_iff_mem_nhds_isVonNBounded, withSeminorms_pi
83
Total91

LinearMap

Theorems

NameKindAssumesProvesValidatesDepends On
withSeminorms_induced πŸ“–mathematicalWithSeminormsSeminormFamily.comp
TopologicalSpace.induced
DFunLike.coe
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
instFunLike
TopologicalSpace
β€”WithSeminorms.topologicalAddGroup
topologicalAddGroup_induced
DistribMulActionSemiHomClass.toAddMonoidHomClass
SemilinearMapClass.distribMulActionSemiHomClass
semilinearMapClass
SeminormFamily.withSeminorms_iff_nhds_eq_iInf
nhds_induced
map_zero
AddMonoidHomClass.toZeroHomClass
Filter.comap_iInf
iInf_congr
Filter.comap_comap

NormedSpace

Theorems

NameKindAssumesProvesValidatesDepends On
toLocallyConvexSpace πŸ“–mathematicalβ€”LocallyConvexSpace
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
toModule
Real.normedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
β€”toLocallyConvexSpace'
IsScalarTower.left
toLocallyConvexSpace' πŸ“–mathematicalβ€”LocallyConvexSpace
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
β€”WithSeminorms.toLocallyConvexSpace
norm_withSeminorms

PolynormableSpace

Theorems

NameKindAssumesProvesValidatesDepends On
iInf πŸ“–mathematicalPolynormableSpaceiInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”WithSeminorms.toPolynormableSpace
withSeminorms_iInf
withSeminorms'
induced πŸ“–mathematicalβ€”PolynormableSpace
TopologicalSpace.induced
DFunLike.coe
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
TopologicalSpace
β€”WithSeminorms.toPolynormableSpace
LinearMap.withSeminorms_induced
withSeminorms
inf πŸ“–mathematicalβ€”PolynormableSpace
TopologicalSpace
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_pair
sInf
sInf πŸ“–mathematicalPolynormableSpaceInfSet.sInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”sInf_eq_iInf'
iInf
withSeminorms πŸ“–mathematicalβ€”WithSeminorms
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Continuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm.instFunLike
β€”withSeminorms'
withSeminorms' πŸ“–mathematicalβ€”WithSeminorms
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Continuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm.instFunLike
β€”β€”

Seminorm

Theorems

NameKindAssumesProvesValidatesDepends On
bound_of_continuous πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
Continuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
instFunLike
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
NNReal
Ring.toSemiring
SeminormedRing.toRing
instSMul
Algebra.toSMul
instCommSemiringNNReal
Real.semiring
NNReal.instAlgebraOfReal
Algebra.id
Real.instCommSemiring
CommSemiring.toSemiring
IsScalarTower.left
instSemiringNNReal
NNReal.instMulActionOfReal
DistribMulAction.toMulAction
Real.instMonoid
AddCommMonoid.toAddMonoid
Real.instAddCommMonoid
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Finset.sup
instSemilatticeSup
instOrderBot
β€”IsScalarTower.left
Filter.HasBasis.mem_iff
WithSeminorms.hasBasis
ball_mem_nhds
one_pos
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
SeminormFamily.basisSets_iff
le_of_eq
SeminormClass.map_smul_eq_mul
instSeminormClass
continuous
SeminormedAddCommGroup.toIsTopologicalAddGroup
UniformContinuousConstSMul.to_continuousConstSMul
IsBoundedSMul.toUniformContinuousConstSMul
NormedSpace.toIsBoundedSMul
Filter.mem_of_superset
Metric.ball_mem_nhds
bound_of_continuous_normedSpace
LT.lt.le
LT.lt.ne
bound_of_continuous_normedSpace πŸ“–mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
instFunLike
Real.instLT
Real.instZero
Real.instLE
Real.instMul
Norm.norm
SeminormedAddCommGroup.toNorm
β€”Continuous.tendsto
map_zero
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
instSeminormClass
Filter.HasBasis.mem_iff
NormedAddCommGroup.nhds_zero_basis_norm_lt
Iio_mem_nhds
instClosedIciTopology
OrderTopology.to_orderClosedTopology
instOrderTopologyReal
one_pos
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
NormedField.exists_one_lt_norm
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
lt_trans
Mathlib.Meta.Positivity.pos_of_isNat
Real.instIsOrderedRing
Real.instNontrivial
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_one
MulZeroClass.mul_zero
le_of_eq
map_eq_zero_of_norm_eq_zero
bound_of_shell
LE.le.trans
le_of_lt
div_le_iffβ‚€'
one_div_div
const_isBounded πŸ“–mathematicalβ€”IsBounded
Seminorm
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comp
NNReal
instSMul
Algebra.toSMul
Real
instCommSemiringNNReal
Real.semiring
NNReal.instAlgebraOfReal
Algebra.id
Real.instCommSemiring
CommSemiring.toSemiring
IsScalarTower.left
instSemiringNNReal
NNReal.instMulActionOfReal
DistribMulAction.toMulAction
Real.instMonoid
AddCommMonoid.toAddMonoid
Real.instAddCommMonoid
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
β€”IsScalarTower.left
le_trans
smul_le_smul
Finset.sup_le
le_rfl
Finset.sup_singleton
cont_normedSpace_to_withSeminorms πŸ“–mathematicalWithSeminorms
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SeminormedAddCommGroup.toAddCommGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NormedSpace.toModule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comp
NNReal
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
instSMul
Algebra.toSMul
Real
instCommSemiringNNReal
Real.semiring
NNReal.instAlgebraOfReal
Algebra.id
Real.instCommSemiring
CommSemiring.toSemiring
IsScalarTower.left
instSemiringNNReal
NNReal.instMulActionOfReal
DistribMulAction.toMulAction
Real.instMonoid
AddCommMonoid.toAddMonoid
Real.instAddCommMonoid
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
normSeminorm
Continuous
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
DFunLike.coe
LinearMap
LinearMap.instFunLike
β€”IsScalarTower.left
continuous_from_bounded
norm_withSeminorms
const_isBounded
cont_withSeminorms_normedSpace πŸ“–mathematicalWithSeminorms
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comp
SeminormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
normSeminorm
NNReal
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
instSMul
Algebra.toSMul
Real
instCommSemiringNNReal
Real.semiring
NNReal.instAlgebraOfReal
Algebra.id
Real.instCommSemiring
CommSemiring.toSemiring
IsScalarTower.left
instSemiringNNReal
NNReal.instMulActionOfReal
DistribMulAction.toMulAction
Real.instMonoid
AddCommMonoid.toAddMonoid
Real.instAddCommMonoid
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Finset.sup
instSemilatticeSup
instOrderBot
Continuous
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
DFunLike.coe
LinearMap
LinearMap.instFunLike
β€”IsScalarTower.left
continuous_from_bounded
norm_withSeminorms
isBounded_const
continuous_from_bounded πŸ“–mathematicalWithSeminorms
IsBounded
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
Continuous
DFunLike.coe
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
β€”WithSeminorms.topologicalAddGroup
continuous_of_continuous_comp
continuous_of_le
IsScalarTower.left
map_sum
AddMonoidHom.instAddMonoidHomClass
Finset.sum_apply
continuous_finset_sum
IsTopologicalSemiring.toContinuousAdd
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Continuous.const_smul
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
WithSeminorms.continuous_seminorm
LE.le.trans
finset_sup_smul
finset_sup_le_sum
continuous_iff_continuous_comp πŸ“–mathematicalWithSeminormsContinuous
DFunLike.coe
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
instFunLike
comp
β€”Continuous.comp
WithSeminorms.continuous_seminorm
continuous_of_continuous_comp
continuous_of_continuous_comp πŸ“–mathematicalWithSeminorms
Continuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
instFunLike
comp
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
LinearMap.instFunLike
β€”WithSeminorms.topologicalAddGroup
continuous_of_continuousAt_zero
IsTopologicalAddGroup.toContinuousAdd
DistribMulActionSemiHomClass.toAddMonoidHomClass
SemilinearMapClass.distribMulActionSemiHomClass
LinearMap.semilinearMapClass
LinearMap.map_zero
SeminormFamily.withSeminorms_iff_nhds_eq_iInf
map_zero
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
instSeminormClass
ContinuousAt.tendsto
Continuous.continuousAt
isBounded_const πŸ“–mathematicalβ€”IsBounded
Seminorm
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comp
NNReal
instSMul
Algebra.toSMul
Real
instCommSemiringNNReal
Real.semiring
NNReal.instAlgebraOfReal
Algebra.id
Real.instCommSemiring
CommSemiring.toSemiring
IsScalarTower.left
instSemiringNNReal
NNReal.instMulActionOfReal
DistribMulAction.toMulAction
Real.instMonoid
AddCommMonoid.toAddMonoid
Real.instAddCommMonoid
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Finset.sup
instSemilatticeSup
instOrderBot
β€”IsScalarTower.left
isBounded_sup πŸ“–mathematicalIsBoundedSeminorm
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comp
Finset.sup
instSemilatticeSup
instOrderBot
NNReal
instSMul
Algebra.toSMul
Real
instCommSemiringNNReal
Real.semiring
NNReal.instAlgebraOfReal
Algebra.id
Real.instCommSemiring
CommSemiring.toSemiring
IsScalarTower.left
instSemiringNNReal
NNReal.instMulActionOfReal
DistribMulAction.toMulAction
Real.instMonoid
AddCommMonoid.toAddMonoid
Real.instAddCommMonoid
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
β€”IsScalarTower.left
Finset.eq_empty_or_nonempty
comp.congr_simp
Finset.sup_empty
zero_comp
smul_zero
LE.le.trans
smul_le_smul
Finset.sup_mono
Finset.subset_biUnion_of_mem
Finset.le_sup
comp_mono
finset_sup_le_sum
map_sum
AddMonoidHom.instAddMonoidHomClass
Finset.sum_congr
Finset.sum_le_sum
IsOrderedAddMonoid.toAddLeftMono
IsOrderedCancelAddMonoid.toIsOrderedAddMonoid
instIsOrderedCancelAddMonoid
Finset.sum_const
smul_assoc
AddCommMonoid.nat_isScalarTower
le_refl
map_eq_zero_of_norm_eq_zero πŸ“–β€”Continuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
instFunLike
Norm.norm
SeminormedAddCommGroup.toNorm
Real.instZero
β€”β€”Specializes.eq
T5Space.toT1Space
OrderTopology.t5Space
instOrderTopologyReal
Specializes.map
specializes_iff_mem_closure
mem_closure_zero_iff_norm
map_zero
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
instSeminormClass
map_eq_zero_of_norm_zero πŸ“–β€”Continuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
instFunLike
Norm.norm
SeminormedAddCommGroup.toNorm
Real.instZero
β€”β€”map_eq_zero_of_norm_eq_zero

SeminormFamily

Definitions

NameCategoryTheorems
addGroupFilterBasis πŸ“–CompOp
2 mathmath: basisSets_smul, basisSets_smul_left
basisSets πŸ“–CompOp
8 mathmath: basisSets_univ_mem, ContinuousLinearMapWOT.hasBasis_seminorms, LinearMap.hasBasis_weakBilin, basisSets_nonempty, basisSets_singleton_mem, WithSeminorms.hasBasis, basisSets_mem, basisSets_iff
comp πŸ“–CompOp
5 mathmath: comp_apply, finset_sup_comp, Topology.IsInducing.withSeminorms, withSeminorms_pi, LinearMap.withSeminorms_induced
moduleFilterBasis πŸ“–CompOp
3 mathmath: filter_eq_iInf, WithSeminorms.withSeminorms_eq, WithSeminorms.topology_eq_withSeminorms
sigma πŸ“–CompOp
2 mathmath: withSeminorms_pi, withSeminorms_iInf

Theorems

NameKindAssumesProvesValidatesDepends On
basisSets_add πŸ“–mathematicalSet
Set.instMembership
basisSets
Set.instHasSubset
Set.add
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
β€”basisSets_iff
Nat.instAtLeastTwoHAddOfNat
basisSets_mem
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
zero_lt_two
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
Set.Subset.trans
Seminorm.ball_add_ball_subset
add_zero
add_halves
CharZero.NeZero.two
subset_refl
Set.instReflSubset
basisSets_iff πŸ“–mathematicalβ€”Set
Set.instMembership
basisSets
Real
Real.instLT
Real.instZero
Seminorm.ball
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Finset.sup
Seminorm
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”β€”
basisSets_intersect πŸ“–mathematicalSet
Set.instMembership
basisSets
Set.instHasSubset
Set.instInter
β€”basisSets_iff
basisSets_mem
lt_min_iff
Seminorm.ball_finset_sup_eq_iInter
Set.subset_inter
Set.iInterβ‚‚_mono'
Finset.subset_union_left
Seminorm.ball_mono
min_le_left
Finset.subset_union_right
min_le_right
basisSets_mem πŸ“–mathematicalReal
Real.instLT
Real.instZero
Set
Set.instMembership
basisSets
Seminorm.ball
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Finset.sup
Seminorm
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”basisSets_iff
basisSets_mem_nhds πŸ“–mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
Set
Set.instMembership
basisSets
Filter
Filter.instMembership
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”basisSets_iff
Seminorm.ball_mem_nhds
Finset.induction_on
Finset.sup_empty
continuous_zero
Finset.sup_insert
Continuous.max
OrderTopology.to_orderClosedTopology
instOrderTopologyReal
basisSets_neg πŸ“–mathematicalSet
Set.instMembership
basisSets
Set.instHasSubset
Set.preimage
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”basisSets_iff
Set.neg_preimage
Seminorm.neg_ball
neg_zero
Eq.subset
Set.instReflSubset
basisSets_nonempty πŸ“–mathematicalβ€”Set.Nonempty
Set
basisSets
β€”Set.nonempty_def
basisSets_univ_mem
basisSets_singleton_mem πŸ“–mathematicalReal
Real.instLT
Real.instZero
Set
Set.instMembership
basisSets
Seminorm.ball
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”basisSets_iff
Finset.sup_singleton
basisSets_smul πŸ“–mathematicalSet
Set.instMembership
basisSets
Filter
Filter.instMembership
nhds
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
FilterBasis.sets
AddGroupFilterBasis.toFilterBasis
AddCommGroup.toAddGroup
addGroupFilterBasis
Set.instHasSubset
Set.smul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
β€”basisSets_iff
Metric.ball_mem_nhds
Real.sqrt_pos
basisSets_mem
Set.Subset.trans
Seminorm.ball_smul_ball
Real.mul_self_sqrt
le_of_lt
subset_refl
Set.instReflSubset
basisSets_smul_left πŸ“–mathematicalSet
Set.instMembership
basisSets
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
FilterBasis.sets
AddGroupFilterBasis.toFilterBasis
AddCommGroup.toAddGroup
addGroupFilterBasis
Set.instHasSubset
Set.preimage
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
β€”basisSets_iff
Seminorm.smul_ball_preimage
smul_zero
basisSets_mem
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
norm_pos_iff
Set.Subset.rfl
not_ne_iff
zero_smul
Set.preimage_const_of_mem
map_zero
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
Seminorm.instSeminormClass
basisSets_smul_right πŸ“–mathematicalSet
Set.instMembership
basisSets
Filter.Eventually
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
nhds
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”basisSets_iff
Filter.eventually_iff
Seminorm.mem_ball_zero
SeminormClass.map_smul_eq_mul
Seminorm.instSeminormClass
lt_div_iffβ‚€
MulPosReflectLE.toMulPosReflectLT
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Real.instIsStrictOrderedRing
ball_zero_eq
Metric.ball_mem_nhds
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
le_antisymm
NonnegHomClass.apply_nonneg
AddGroupSeminormClass.toNonnegHomClass
Real.instIsOrderedAddMonoid
SeminormClass.toAddGroupSeminormClass
MulZeroClass.mul_zero
IsOpen.mem_nhds
isOpen_univ
Set.mem_univ
basisSets_univ_mem πŸ“–mathematicalβ€”Set
Set.instMembership
basisSets
Set.univ
β€”basisSets_iff
one_pos
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
Finset.sup_empty
Seminorm.bot_eq_zero
Seminorm.ball_zero'
basisSets_zero πŸ“–mathematicalSet
Set.instMembership
basisSets
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”basisSets_iff
Seminorm.mem_ball_zero
map_zero
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
Seminorm.instSeminormClass
comp_apply πŸ“–mathematicalβ€”comp
Seminorm.comp
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
β€”β€”
filter_eq_iInf πŸ“–mathematicalβ€”FilterBasis.filter
AddGroupFilterBasis.toFilterBasis
AddCommGroup.toAddGroup
ModuleFilterBasis.toAddGroupFilterBasis
Field.toCommRing
NormedField.toField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
moduleFilterBasis
iInf
Filter
Filter.instInfSet
Filter.comap
Real
DFunLike.coe
Seminorm
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
nhds
Real.pseudoMetricSpace
Real.instZero
β€”le_antisymm
le_iInf
Filter.HasBasis.le_basis_iff
FilterBasis.hasBasis
Filter.HasBasis.comap
Metric.nhds_basis_ball
Finset.sup_singleton
basisSets_mem
Seminorm.ball_zero_eq_preimage_ball
subset_refl
Set.instReflSubset
Filter.HasBasis.ge_iff
basisSets_iff
Seminorm.ball_finset_sup_eq_iInter
Finset.iInter_mem_sets
Filter.mem_iInf_of_mem
Metric.ball_mem_nhds
Eq.subset
finset_sup_comp πŸ“–mathematicalβ€”Seminorm.comp
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
Finset.sup
Seminorm
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
comp
β€”Seminorm.ext
Seminorm.comp_apply
NonnegHomClass.apply_nonneg
AddGroupSeminormClass.toNonnegHomClass
Real.instIsOrderedAddMonoid
SeminormClass.toAddGroupSeminormClass
Seminorm.instSeminormClass
Seminorm.finset_sup_apply
withSeminorms_iff_nhds_eq_iInf πŸ“–mathematicalβ€”WithSeminorms
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
iInf
Filter
Filter.instInfSet
Filter.comap
Real
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instZero
β€”filter_eq_iInf
WithSeminorms.topology_eq_withSeminorms
AddGroupFilterBasis.nhds_zero_eq
withSeminorms_of_nhds
withSeminorms_iff_topologicalSpace_eq_iInf πŸ“–mathematicalβ€”WithSeminorms
iInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
AddGroupSeminorm.toSeminormedAddCommGroup
Seminorm.toAddGroupSeminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
β€”withSeminorms_iff_nhds_eq_iInf
IsTopologicalAddGroup.ext_iff
topologicalAddGroup_iInf
SeminormedAddCommGroup.toIsTopologicalAddGroup
nhds_iInf
comap_norm_nhds_zero
withSeminorms_iff_uniformSpace_eq_iInf πŸ“–mathematicalβ€”WithSeminorms
UniformSpace.toTopologicalSpace
iInf
UniformSpace
instInfSetUniformSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
AddGroupSeminorm.toSeminormedAddCommGroup
Seminorm.toAddGroupSeminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
β€”withSeminorms_iff_nhds_eq_iInf
IsUniformAddGroup.to_topologicalAddGroup
IsUniformAddGroup.ext_iff
isUniformAddGroup_iInf
SeminormedAddCommGroup.to_isUniformAddGroup
UniformSpace.toTopologicalSpace_iInf
nhds_iInf
comap_norm_nhds_zero
withSeminorms_of_hasBasis πŸ“–mathematicalFilter.HasBasis
Set
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Set.instMembership
basisSets
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
WithSeminormsβ€”withSeminorms_of_nhds
Filter.HasBasis.eq_of_same_basis
FilterBasis.hasBasis
withSeminorms_of_nhds πŸ“–mathematicalnhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
FilterBasis.filter
AddGroupFilterBasis.toFilterBasis
AddCommGroup.toAddGroup
ModuleFilterBasis.toAddGroupFilterBasis
Field.toCommRing
NormedField.toField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
moduleFilterBasis
WithSeminormsβ€”IsTopologicalAddGroup.ext
AddGroupFilterBasis.isTopologicalAddGroup
AddGroupFilterBasis.nhds_zero_eq

Topology.IsInducing

Theorems

NameKindAssumesProvesValidatesDepends On
polynormableSpace πŸ“–mathematicalTopology.IsInducing
DFunLike.coe
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
PolynormableSpaceβ€”WithSeminorms.toPolynormableSpace
withSeminorms
PolynormableSpace.withSeminorms
withSeminorms πŸ“–mathematicalWithSeminorms
Topology.IsInducing
DFunLike.coe
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
SeminormFamily.compβ€”eq_induced
LinearMap.withSeminorms_induced

WithSeminorms

Theorems

NameKindAssumesProvesValidatesDepends On
T1_of_separating πŸ“–mathematicalWithSeminormsT1Spaceβ€”topologicalAddGroup
IsTopologicalAddGroup.t1Space
IsTopologicalAddGroup.toContinuousAdd
isOpen_compl_iff
isOpen_iff_mem_balls
lt_of_le_of_ne'
NonnegHomClass.apply_nonneg
AddGroupSeminormClass.toNonnegHomClass
Real.instIsOrderedAddMonoid
SeminormClass.toAddGroupSeminormClass
Seminorm.instSeminormClass
Set.subset_compl_singleton_iff
Finset.sup_singleton
Seminorm.mem_ball
zero_sub
AddGroupSeminormClass.map_neg_eq_map
not_lt
le_refl
congr_equiv πŸ“–mathematicalWithSeminormsSeminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”congr
RingHomIsometric.ids
Seminorm.comp_id
Finset.sup_singleton
one_smul
IsScalarTower.left
Equiv.apply_symm_apply
continuousSMul πŸ“–mathematicalWithSeminormsContinuousSMul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
β€”withSeminorms_eq
ModuleFilterBasis.continuousSMul
IsTopologicalDivisionRing.toIsTopologicalRing
NormedDivisionRing.to_isTopologicalDivisionRing
continuous_seminorm πŸ“–mathematicalWithSeminormsContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
β€”topologicalAddGroup
SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf
continuous_iInf_dom
continuous_norm
equicontinuous_TFAE πŸ“–mathematicalWithSeminorms
UniformSpace.toTopologicalSpace
List.TFAE
EquicontinuousAt
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
NontriviallyNormedField.toNormedField
AddCommGroup.toAddCommMonoid
DFunLike.coe
LinearMap.instFunLike
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Equicontinuous
UniformEquicontinuous
β€”SeminormFamily.withSeminorms_iff_uniformSpace_eq_iInf
uniformEquicontinuous_iInf_rng
equicontinuous_iInf_rng
equicontinuousAt_iInf_rng
List.forall_tfae
uniformEquicontinuous_of_equicontinuousAt_zero
SeminormedAddCommGroup.to_isUniformAddGroup
DistribMulActionSemiHomClass.toAddMonoidHomClass
SemilinearMapClass.distribMulActionSemiHomClass
LinearMap.semilinearMapClass
UniformEquicontinuous.equicontinuous
Filter.mp_mem
Metric.equicontinuousAt_iff_right
one_pos
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
Filter.univ_mem'
map_zero
AddMonoidHomClass.toZeroHomClass
dist_zero
LT.lt.le
Seminorm.bddAbove_of_absorbent
absorbent_nhds_zero
Set.forall_mem_range
Seminorm.coe_iSup_eq
Seminorm.continuous'
IsUniformAddGroup.to_topologicalAddGroup
ContinuousSMul.continuousConstSMul
Seminorm.closedBall_iSup
le_ciSup
Metric.equicontinuousAt_of_continuity_modulus
Continuous.tendsto
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
Seminorm.instSeminormClass
Filter.Eventually.of_forall
List.tfae_of_cycle
finset_sups πŸ“–mathematicalWithSeminormsFinset
Finset.sup
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”congr
RingHomIsometric.ids
IsScalarTower.left
one_smul
le_refl
Finset.sup_singleton
firstCountableTopology πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
FirstCountableTopologyβ€”topologicalAddGroup
isUniformAddGroup_of_addCommGroup
SeminormFamily.withSeminorms_iff_nhds_eq_iInf
Filter.iInf.isCountablyGenerated
Filter.comap.isCountablyGenerated
FirstCountableTopology.nhds_generated_countable
TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology
PseudoEMetricSpace.pseudoMetrizableSpace
IsUniformAddGroup.uniformity_countably_generated
UniformSpace.firstCountableTopology
hasBasis πŸ“–mathematicalWithSeminormsFilter.HasBasis
Set
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Set.instMembership
SeminormFamily.basisSets
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
β€”topology_eq_withSeminorms
AddGroupFilterBasis.nhds_zero_hasBasis
hasBasis_ball πŸ“–mathematicalWithSeminormsFilter.HasBasis
Finset
Real
nhds
Real.instLT
Real.instZero
Seminorm.ball
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Finset.sup
Seminorm
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”topologicalAddGroup
map_add_left_nhds_zero
Seminorm.vadd_ball
add_zero
vadd_eq_add
Filter.HasBasis.map
hasBasis_zero_ball
hasBasis_zero_ball πŸ“–mathematicalWithSeminormsFilter.HasBasis
Finset
Real
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Real.instLT
Real.instZero
Seminorm.ball
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Finset.sup
Seminorm
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”Filter.HasBasis.mem_iff
hasBasis
image_isVonNBounded_iff_finset_seminorm_bounded πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
Bornology.IsVonNBounded
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Set.image
Real
Real.instLT
Real.instZero
DFunLike.coe
Seminorm
Ring.toSemiring
SeminormedRing.toRing
Seminorm.instFunLike
Finset.sup
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”isVonNBounded_iff_finset_seminorm_bounded
image_isVonNBounded_iff_seminorm_bounded πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
Bornology.IsVonNBounded
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Set.image
Real
Real.instLT
Real.instZero
DFunLike.coe
Seminorm
Ring.toSemiring
SeminormedRing.toRing
Seminorm.instFunLike
β€”isVonNBounded_iff_seminorm_bounded
isOpen_iff_mem_balls πŸ“–mathematicalWithSeminormsIsOpen
Real
Real.instLT
Real.instZero
Set
Set.instHasSubset
Seminorm.ball
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Finset.sup
Seminorm
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”mem_nhds_iff
isVonNBounded_iff_finset_seminorm_bounded πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
Bornology.IsVonNBounded
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Real
Real.instLT
Real.instZero
DFunLike.coe
Seminorm
Ring.toSemiring
SeminormedRing.toRing
Seminorm.instFunLike
Finset.sup
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”Filter.HasBasis.isVonNBounded_iff
hasBasis
Absorbs.exists_pos
SeminormFamily.basisSets_mem
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
NormedField.exists_lt_norm
lt_trans
Seminorm.mem_ball_zero
mul_one
Seminorm.smul_ball_zero
norm_pos_iff
LT.lt.trans
le_of_lt
SeminormFamily.basisSets_iff
Absorbs.mono_right
Seminorm.ball_zero_absorbs_ball_zero
isVonNBounded_iff_seminorm_bddAbove πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
Bornology.IsVonNBounded
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
BddAbove
Real
Real.instLE
Set.image
DFunLike.coe
Seminorm
Ring.toSemiring
SeminormedRing.toRing
Seminorm.instFunLike
β€”isVonNBounded_iff_seminorm_bounded
isVonNBounded_iff_seminorm_bounded πŸ“–mathematicalWithSeminorms
NontriviallyNormedField.toNormedField
Bornology.IsVonNBounded
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Real
Real.instLT
Real.instZero
DFunLike.coe
Seminorm
Ring.toSemiring
SeminormedRing.toRing
Seminorm.instFunLike
β€”isVonNBounded_iff_finset_seminorm_bounded
Finset.sup_singleton
lt_of_lt_of_le
Finset.le_sup'
Seminorm.finset_sup_apply_lt
Eq.le
Finset.sup_empty
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
mem_nhds_iff πŸ“–mathematicalWithSeminormsSet
Filter
Filter.instMembership
nhds
Real
Real.instLT
Real.instZero
Set.instHasSubset
Seminorm.ball
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Finset.sup
Seminorm
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
β€”Filter.HasBasis.mem_iff
hasBasis_ball
partial_sups πŸ“–mathematicalWithSeminormsFinset.sup
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
Finset.Iic
β€”congr
RingHomIsometric.ids
IsScalarTower.left
one_smul
le_refl
Finset.sup_singleton
Finset.le_sup
Finset.mem_Iic
le_rfl
separating_iff_T1 πŸ“–mathematicalWithSeminormsT1Spaceβ€”T1_of_separating
separating_of_T1
separating_of_T1 πŸ“–β€”WithSeminormsβ€”β€”List.TFAE.out
t1Space_TFAE
Filter.HasBasis.specializes_iff
hasBasis_zero_ball
Seminorm.ball_finset_sup_eq_iInter
tendsto_nhds πŸ“–mathematicalWithSeminormsFilter.Tendsto
nhds
Filter.Eventually
Real
Real.instLT
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
SubNegMonoid.toSub
β€”tendsto_nhds'
Finset.sup_singleton
Filter.Eventually.mono
Finset.eventually_all
Seminorm.finset_sup_apply_lt
tendsto_nhds' πŸ“–mathematicalWithSeminormsFilter.Tendsto
nhds
Filter.Eventually
Real
Real.instLT
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
Finset.sup
Seminorm.instSemilatticeSup
Seminorm.instOrderBot
SubNegMonoid.toSub
β€”Filter.HasBasis.tendsto_right_iff
hasBasis_ball
tendsto_nhds_atTop πŸ“–mathematicalWithSeminormsFilter.Tendsto
Filter.atTop
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
nhds
Real
Real.instLT
DFunLike.coe
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Seminorm.instFunLike
SubNegMonoid.toSub
β€”tendsto_nhds
Filter.eventually_atTop
SemilatticeSup.instIsDirectedOrder
toLocallyConvexSpace πŸ“–mathematicalWithSeminormsLocallyConvexSpace
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
β€”topologicalAddGroup
LocallyConvexSpace.ofBasisZero
topology_eq_withSeminorms
AddGroupFilterBasis.nhds_eq
AddGroupFilterBasis.N_zero
FilterBasis.hasBasis
Seminorm.convex_ball
toPolynormableSpace πŸ“–mathematicalWithSeminormsPolynormableSpaceβ€”topologicalAddGroup
continuous_seminorm
SeminormFamily.withSeminorms_iff_nhds_eq_iInf
le_antisymm
Continuous.tendsto'
map_zero
AddGroupSeminormClass.toZeroHomClass
SeminormClass.toAddGroupSeminormClass
Seminorm.instSeminormClass
iInf_le
topologicalAddGroup πŸ“–mathematicalWithSeminormsIsTopologicalAddGroup
AddCommGroup.toAddGroup
β€”withSeminorms_eq
AddGroupFilterBasis.isTopologicalAddGroup
topology_eq_withSeminorms πŸ“–mathematicalWithSeminormsModuleFilterBasis.topology
Field.toCommRing
NormedField.toField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SeminormFamily.moduleFilterBasis
β€”β€”
uniformEquicontinuous_iff_bddAbove_and_continuous_iSup πŸ“–mathematicalWithSeminorms
UniformSpace.toTopologicalSpace
UniformEquicontinuous
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
NontriviallyNormedField.toNormedField
AddCommGroup.toAddCommMonoid
DFunLike.coe
LinearMap.instFunLike
BddAbove
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Module.toDistribMulAction
Preorder.toLE
PartialOrder.toPreorder
Seminorm.instPartialOrder
Set.range
Seminorm.comp
Continuous
Real
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
iSup
Pi.supSet
Real.instSupSet
Seminorm.instFunLike
β€”List.TFAE.out
equicontinuous_TFAE
uniformEquicontinuous_iff_exists_continuous_seminorm πŸ“–mathematicalWithSeminorms
UniformSpace.toTopologicalSpace
UniformEquicontinuous
LinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
NontriviallyNormedField.toNormedField
AddCommGroup.toAddCommMonoid
DFunLike.coe
LinearMap.instFunLike
Continuous
Real
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Seminorm
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
AddCommGroup.toAddGroup
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Module.toDistribMulAction
Seminorm.instFunLike
Ring.toSemiring
SeminormedRing.toRing
Preorder.toLE
PartialOrder.toPreorder
Seminorm.instPartialOrder
Seminorm.comp
β€”List.TFAE.out
equicontinuous_TFAE
withSeminorms_eq πŸ“–mathematicalWithSeminormsModuleFilterBasis.topology
Field.toCommRing
NormedField.toField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SeminormFamily.moduleFilterBasis
β€”topology_eq_withSeminorms

(root)

Definitions

NameCategoryTheorems
PolynormableSpace πŸ“–CompData
8 mathmath: instPolynormableSpaceProd, instPolynormableSpaceSubtypeMemSubmodule, instPolynormableSpace, PolynormableSpace.induced, LocallyConvexSpace.toPolynormableSpace, Topology.IsInducing.polynormableSpace, WithSeminorms.toPolynormableSpace, PolynormableSpace.inf
SeminormFamily πŸ“–CompOpβ€”
WithSeminorms πŸ“–CompData
16 mathmath: withSeminorms_iff_mem_nhds_isVonNBounded, schwartz_withSeminorms, ContinuousLinearMapWOT.withSeminorms, SeminormFamily.withSeminorms_of_hasBasis, SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf, with_gaugeSeminormFamily, SeminormFamily.withSeminorms_iff_uniformSpace_eq_iInf, LinearMap.weakBilin_withSeminorms, PolynormableSpace.withSeminorms', SeminormFamily.withSeminorms_iff_nhds_eq_iInf, norm_withSeminorms, ContDiffMapSupportedIn.withSeminorms, PointwiseConvergenceCLM.withSeminorms, ContDiffMapSupportedIn.withSeminorms', PolynormableSpace.withSeminorms, SeminormFamily.withSeminorms_of_nhds

Theorems

NameKindAssumesProvesValidatesDepends On
instLocallyConvexSpaceRealOfPolynormableSpace πŸ“–mathematicalβ€”LocallyConvexSpace
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
β€”WithSeminorms.toLocallyConvexSpace
IsScalarTower.left
PolynormableSpace.withSeminorms
instPolynormableSpace πŸ“–mathematicalβ€”PolynormableSpace
SeminormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
β€”WithSeminorms.toPolynormableSpace
norm_withSeminorms
instPolynormableSpaceForall πŸ“–mathematicalPolynormableSpacePi.addCommGroup
Pi.module
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
Pi.topologicalSpace
β€”PolynormableSpace.iInf
PolynormableSpace.induced
RingHomIsometric.ids
instPolynormableSpaceProd πŸ“–mathematicalβ€”PolynormableSpace
Prod.instAddCommGroup
Prod.instModule
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
instTopologicalSpaceProd
β€”PolynormableSpace.inf
PolynormableSpace.induced
RingHomIsometric.ids
instPolynormableSpaceSubtypeMemSubmodule πŸ“–mathematicalβ€”PolynormableSpace
Submodule
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
SetLike.instMembership
Submodule.setLike
Submodule.addCommGroup
NormedRing.toRing
NormedCommRing.toNormedRing
NormedField.toNormedCommRing
Submodule.module
instTopologicalSpaceSubtype
β€”Topology.IsInducing.polynormableSpace
RingHomIsometric.ids
Topology.IsInducing.subtypeVal
norm_withSeminorms πŸ“–mathematicalβ€”WithSeminorms
SeminormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
normSeminorm
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
β€”SeminormFamily.withSeminorms_iff_nhds_eq_iInf
SeminormedAddCommGroup.toIsTopologicalAddGroup
iInf_const
coe_normSeminorm
comap_norm_nhds_zero
withSeminorms_iInf πŸ“–mathematicalWithSeminormsSeminormFamily.sigma
iInf
TopologicalSpace
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
TopologicalSpace.instCompleteLattice
β€”WithSeminorms.topologicalAddGroup
topologicalAddGroup_iInf
SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf
iInf_sigma
iInf_congr
withSeminorms_iff_mem_nhds_isVonNBounded πŸ“–mathematicalβ€”WithSeminorms
NontriviallyNormedField.toNormedField
Set
Filter
Filter.instMembership
nhds
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Seminorm.ball
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
Module.toDistribMulAction
AddCommGroup.toAddCommMonoid
Real
Real.instOne
Bornology.IsVonNBounded
β€”WithSeminorms.mem_nhds_iff
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
Finset.univ_unique
Finset.sup_singleton
Set.instReflSubset
WithSeminorms.isVonNBounded_iff_seminorm_bounded
sub_zero
SeminormFamily.withSeminorms_of_nhds
Filter.ext
Filter.Eventually.exists
NontriviallyNormedField.cobounded_neBot
Filter.Eventually.and
Bornology.eventually_ne_cobounded
norm_inv
Seminorm.smul_ball_zero
mul_inv_cancelβ‚€
Set.smul_set_subset_smul_set_iffβ‚€
Filter.mem_of_superset
FilterBasis.mem_filter_of_mem
SeminormFamily.basisSets_singleton_mem
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
FilterBasis.mem_filter_iff
SeminormFamily.basisSets_iff
eq_or_ne
Finset.sup_empty
Finset.nonempty_of_ne_empty
Finset.ext
NormedField.exists_norm_lt
Seminorm.ball_mono
mul_one
LT.lt.le
smul_zero
smul_mem_nhds_smulβ‚€
withSeminorms_pi πŸ“–mathematicalWithSeminormsPi.addCommGroup
Pi.module
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
AddCommGroup.toAddCommMonoid
SeminormFamily.sigma
SeminormFamily.comp
RingHom.id
Semiring.toNonAssocSemiring
RingHomIsometric.ids
SeminormedCommRing.toSeminormedRing
NormedCommRing.toSeminormedCommRing
NormedField.toNormedCommRing
LinearMap.proj
Pi.topologicalSpace
β€”withSeminorms_iInf
RingHomIsometric.ids
LinearMap.withSeminorms_induced

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