File: Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean

full_name: Filter.ZeroAtFilter
code: /-- If `l` is a filter on `α`, then a function `f : α → β` is `ZeroAtFilter l`
  if it tends to zero along `l`. -/
def ZeroAtFilter [Zero β] [TopologicalSpace β] (l : Filter α) (f : α → β) : Prop 
goal: Prop

full_name: Filter.zero_zeroAtFilter
code: theorem zero_zeroAtFilter [Zero β] [TopologicalSpace β] (l : Filter α) :
    ZeroAtFilter l (0 : α → β) 
goal: l.ZeroAtFilter 0

full_name: Filter.ZeroAtFilter.add
code: nonrec theorem ZeroAtFilter.add [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β]
    {l : Filter α} {f g : α → β} (hf : ZeroAtFilter l f) (hg : ZeroAtFilter l g) :
    ZeroAtFilter l (f + g) 
goal: l.ZeroAtFilter (f + g)

full_name: Filter.ZeroAtFilter.neg
code: nonrec theorem ZeroAtFilter.neg [TopologicalSpace β] [AddGroup β] [ContinuousNeg β] {l : Filter α}
    {f : α → β} (hf : ZeroAtFilter l f) : ZeroAtFilter l (-f) 
goal: l.ZeroAtFilter (-f)

full_name: Filter.ZeroAtFilter.smul
code: theorem ZeroAtFilter.smul [TopologicalSpace β] [Zero 𝕜] [Zero β]
    [SMulWithZero 𝕜 β] [ContinuousConstSMul 𝕜 β] {l : Filter α} {f : α → β} (c : 𝕜)
    (hf : ZeroAtFilter l f) : ZeroAtFilter l (c • f) 
goal: l.ZeroAtFilter (c • f)

full_name: Filter.zeroAtFilterSubmodule
code: /-- `zeroAtFilterSubmodule l` is the submodule of `f : α → β` which
tend to zero along `l`. -/
def zeroAtFilterSubmodule
    [TopologicalSpace β] [Semiring 𝕜] [AddCommMonoid β] [Module 𝕜 β]
    [ContinuousAdd β] [ContinuousConstSMul 𝕜 β]
    (l : Filter α) : Submodule 𝕜 (α → β) 
goal: Submodule 𝕜 (α → β)

full_name: Filter.zeroAtFilterAddSubmonoid
code: /-- `zeroAtFilterAddSubmonoid l` is the additive submonoid of `f : α → β`
which tend to zero along `l`. -/
def zeroAtFilterAddSubmonoid [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β]
    (l : Filter α) : AddSubmonoid (α → β) 
goal: AddSubmonoid (α → β)

full_name: Filter.BoundedAtFilter
code: /-- If `l` is a filter on `α`, then a function `f: α → β` is `BoundedAtFilter l`
if `f =O[l] 1`. -/
def BoundedAtFilter [Norm β] (l : Filter α) (f : α → β) : Prop 
goal: Prop

full_name: Filter.ZeroAtFilter.boundedAtFilter
code: theorem ZeroAtFilter.boundedAtFilter [NormedAddCommGroup β] {l : Filter α} {f : α → β}
    (hf : ZeroAtFilter l f) : BoundedAtFilter l f 
goal: l.BoundedAtFilter f

full_name: Filter.const_boundedAtFilter
code: theorem const_boundedAtFilter [Norm β] (l : Filter α) (c : β) :
    BoundedAtFilter l (Function.const α c : α → β) 
goal: l.BoundedAtFilter (Function.const α c)

full_name: Filter.BoundedAtFilter.add
code: nonrec theorem BoundedAtFilter.add [SeminormedAddCommGroup β] {l : Filter α} {f g : α → β}
    (hf : BoundedAtFilter l f) (hg : BoundedAtFilter l g) : BoundedAtFilter l (f + g) 
goal: l.BoundedAtFilter (f + g)

full_name: Filter.BoundedAtFilter.neg
code: theorem BoundedAtFilter.neg [SeminormedAddCommGroup β] {l : Filter α} {f : α → β}
    (hf : BoundedAtFilter l f) : BoundedAtFilter l (-f) 
goal: l.BoundedAtFilter (-f)

full_name: Filter.BoundedAtFilter.smul
code: theorem BoundedAtFilter.smul
    [SeminormedRing 𝕜] [SeminormedAddCommGroup β] [Module 𝕜 β] [BoundedSMul 𝕜 β]
    {l : Filter α} {f : α → β} (c : 𝕜) (hf : BoundedAtFilter l f) : BoundedAtFilter l (c • f) 
goal: l.BoundedAtFilter (c • f)

full_name: Filter.BoundedAtFilter.mul
code: nonrec theorem BoundedAtFilter.mul [SeminormedRing β] {l : Filter α} {f g : α → β}
    (hf : BoundedAtFilter l f) (hg : BoundedAtFilter l g) : BoundedAtFilter l (f * g) 
goal: l.BoundedAtFilter (f * g)

full_name: Filter.boundedFilterSubmodule
code: /-- The submodule of functions that are bounded along a filter `l`. -/
def boundedFilterSubmodule
    [SeminormedRing 𝕜] [SeminormedAddCommGroup β] [Module 𝕜 β] [BoundedSMul 𝕜 β] (l : Filter α) :
    Submodule 𝕜 (α → β) 
goal: Submodule 𝕜 (α → β)

full_name: Filter.boundedFilterSubalgebra
code: /-- The subalgebra of functions that are bounded along a filter `l`. -/
def boundedFilterSubalgebra
    [SeminormedCommRing 𝕜] [SeminormedRing β] [Algebra 𝕜 β] [BoundedSMul 𝕜 β] (l : Filter α) :
    Subalgebra 𝕜 (α → β) 
goal: Subalgebra 𝕜 (α → β)