File: Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean

full_name: ContinuousMultilinearMap.toUniformOnFun
code: /-- An auxiliary definition used to define topology on `ContinuousMultilinearMap 𝕜 E F`. -/
def toUniformOnFun [TopologicalSpace F] (f : ContinuousMultilinearMap 𝕜 E F) :
    (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F 
goal: ((i : ι) → E i) →ᵤ[{s | Bornology.IsVonNBounded 𝕜 s}] F

full_name: ContinuousMultilinearMap.range_toUniformOnFun
code: lemma range_toUniformOnFun [DecidableEq ι] [TopologicalSpace F] :
    range toUniformOnFun =
      {f : (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F |
        Continuous (toFun _ f) ∧
        (∀ (m : Π i, E i) i x y,
          toFun _ f (update m i (x + y)) = toFun _ f (update m i x) + toFun _ f (update m i y)) ∧
        (∀ (m : Π i, E i) i (c : 𝕜) x,
          toFun _ f (update m i (c • x)) = c • toFun _ f (update m i x))} 
goal: Set.range ContinuousMultilinearMap.toUniformOnFun =
  {f |
    Continuous ((UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f) ∧
      (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
          (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i (x + y)) =
            (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i x) +
              (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i y)) ∧
        ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
          (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i (c • x)) =
            c • (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i x)}

full_name: ContinuousMultilinearMap.toUniformOnFun_toFun
code: @[simp]
lemma toUniformOnFun_toFun [TopologicalSpace F] (f : ContinuousMultilinearMap 𝕜 E F) :
    UniformOnFun.toFun _ f.toUniformOnFun = f 
goal: (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f.toUniformOnFun = ⇑f

full_name: ContinuousMultilinearMap.instTopologicalSpace
code: instance instTopologicalSpace [TopologicalSpace F] [TopologicalAddGroup F] :
    TopologicalSpace (ContinuousMultilinearMap 𝕜 E F) 
goal: TopologicalSpace (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.instUniformSpace
code: instance instUniformSpace [UniformSpace F] [UniformAddGroup F] :
    UniformSpace (ContinuousMultilinearMap 𝕜 E F) 
goal: UniformSpace (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.uniformEmbedding_toUniformOnFun
code: lemma uniformEmbedding_toUniformOnFun :
    UniformEmbedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) 
goal: UniformEmbedding ContinuousMultilinearMap.toUniformOnFun

full_name: ContinuousMultilinearMap.embedding_toUniformOnFun
code: lemma embedding_toUniformOnFun : Embedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) 
goal: Embedding ContinuousMultilinearMap.toUniformOnFun

full_name: ContinuousMultilinearMap.uniformContinuous_coe_fun
code: theorem uniformContinuous_coe_fun [∀ i, ContinuousSMul 𝕜 (E i)] :
    UniformContinuous (DFunLike.coe : ContinuousMultilinearMap 𝕜 E F → (Π i, E i) → F) 
goal: UniformContinuous DFunLike.coe

full_name: ContinuousMultilinearMap.uniformContinuous_eval_const
code: theorem uniformContinuous_eval_const [∀ i, ContinuousSMul 𝕜 (E i)] (x : Π i, E i) :
    UniformContinuous fun f : ContinuousMultilinearMap 𝕜 E F ↦ f x 
goal: UniformContinuous fun f => f x

full_name: ContinuousMultilinearMap.instUniformAddGroup
code: instance instUniformAddGroup : UniformAddGroup (ContinuousMultilinearMap 𝕜 E F) 
goal: UniformAddGroup (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.instUniformContinuousConstSMul
code: instance instUniformContinuousConstSMul {M : Type*}
    [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
    UniformContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F) 
goal: UniformContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.completeSpace
code: theorem completeSpace (h : RestrictGenTopology {s : Set (Π i, E i) | IsVonNBounded 𝕜 s}) :
    CompleteSpace (ContinuousMultilinearMap 𝕜 E F) 
goal: CompleteSpace (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.instCompleteSpace
code: instance instCompleteSpace [∀ i, TopologicalAddGroup (E i)] [SequentialSpace (Π i, E i)] :
    CompleteSpace (ContinuousMultilinearMap 𝕜 E F) 
goal: CompleteSpace (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.instContinuousConstSMul
code: instance instContinuousConstSMul
    {M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
    ContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F) 
goal: ContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.instContinuousSMul
code: instance instContinuousSMul [ContinuousSMul 𝕜 F] :
    ContinuousSMul 𝕜 (ContinuousMultilinearMap 𝕜 E F) 
goal: ContinuousSMul 𝕜 (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis
code: theorem hasBasis_nhds_zero_of_basis {ι : Type*} {p : ι → Prop} {b : ι → Set F}
    (h : (𝓝 (0 : F)).HasBasis p b) :
    (𝓝 (0 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
      (fun Si : Set (Π i, E i) × ι => IsVonNBounded 𝕜 Si.1 ∧ p Si.2)
      fun Si => { f | MapsTo f Si.1 (b Si.2) } 
goal: ∀ {ι_1 : Type u_5} {p : ι_1 → Prop} {b : ι_1 → Set F},
  (𝓝 0).HasBasis p b →
    (𝓝 0).HasBasis (fun Si => Bornology.IsVonNBounded 𝕜 Si.1 ∧ p Si.2) fun Si => {f | Set.MapsTo (⇑f) Si.1 (b Si.2)}

full_name: ContinuousMultilinearMap.hasBasis_nhds_zero
code: theorem hasBasis_nhds_zero :
    (𝓝 (0 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
      (fun SV : Set (Π i, E i) × Set F => IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0) fun SV =>
      { f | MapsTo f SV.1 SV.2 } 
goal: (𝓝 0).HasBasis (fun SV => Bornology.IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0) fun SV => {f | Set.MapsTo (⇑f) SV.1 SV.2}

full_name: ContinuousMultilinearMap.continuous_eval_const
code: theorem continuous_eval_const (x : Π i, E i) :
    Continuous fun p : ContinuousMultilinearMap 𝕜 E F ↦ p x 
goal: Continuous fun p => p x

full_name: ContinuousMultilinearMap.continuous_coe_fun
code: theorem continuous_coe_fun :
    Continuous (DFunLike.coe : ContinuousMultilinearMap 𝕜 E F → (Π i, E i) → F) 
goal: Continuous DFunLike.coe

full_name: ContinuousMultilinearMap.instT2Space
code: instance instT2Space [T2Space F] : T2Space (ContinuousMultilinearMap 𝕜 E F) 
goal: T2Space (ContinuousMultilinearMap 𝕜 E F)

full_name: ContinuousMultilinearMap.apply
code: /-- The application of a multilinear map as a `ContinuousLinearMap`. -/
def apply [ContinuousConstSMul 𝕜 F] (m : Π i, E i) : ContinuousMultilinearMap 𝕜 E F →L[𝕜] F 
goal: ContinuousMultilinearMap 𝕜 E F →L[𝕜] F

full_name: ContinuousMultilinearMap.apply_apply
code: @[simp]
lemma apply_apply [ContinuousConstSMul 𝕜 F] {m : Π i, E i} {c : ContinuousMultilinearMap 𝕜 E F} :
    apply 𝕜 E F m c = c m 
goal: (ContinuousMultilinearMap.apply 𝕜 E F m) c = c m

full_name: ContinuousMultilinearMap.hasSum_eval
code: theorem hasSum_eval {α : Type*} {p : α → ContinuousMultilinearMap 𝕜 E F}
    {q : ContinuousMultilinearMap 𝕜 E F} (h : HasSum p q) (m : Π i, E i) :
    HasSum (fun a => p a m) (q m) 
goal: HasSum (fun a => (p a) m) (q m)

full_name: ContinuousMultilinearMap.tsum_eval
code: theorem tsum_eval [T2Space F] {α : Type*} {p : α → ContinuousMultilinearMap 𝕜 E F} (hp : Summable p)
    (m : Π i, E i) : (∑' a, p a) m = ∑' a, p a m 
goal: (∑' (a : α), p a) m = ∑' (a : α), (p a) m