Documentation

Mathlib.Topology.Instances.NNReal

Topology on ℝ≥0 #

The natural topology on ℝ≥0 (the one induced from ), and a basic API.

Main definitions #

Instances for the following typeclasses are defined:

Everything is inherited from the corresponding structures on the reals.

Main statements #

Various mathematically trivial lemmas are proved about the compatibility of limits and sums in ℝ≥0 and . For example

says that the limit of a filter along a map to ℝ≥0 is the same in and ℝ≥0, and

says that says that a sum of elements in ℝ≥0 is the same in and ℝ≥0.

Similarly, some mathematically trivial lemmas about infinite sums are proved, a few of which rely on the fact that subtraction is continuous.

Real.toNNReal bundled as a continuous map for convenience.

Equations
Instances For
    theorem ContinuousOn.ofReal_map_toNNReal {f : NNRealNNReal} {s : Set } {t : Set NNReal} (hf : ContinuousOn f t) (h : Set.MapsTo Real.toNNReal s t) :
    ContinuousOn (fun (x : ) => (f x.toNNReal)) s

    Embedding of ℝ≥0 to as a bundled continuous map.

    Equations
    Instances For
      instance NNReal.ContinuousMap.canLift {X : Type u_2} [TopologicalSpace X] :
      CanLift C(X, ) C(X, NNReal) ContinuousMap.coeNNRealReal.comp fun (f : C(X, )) => ∀ (x : X), 0 f x
      Equations
      • =
      @[simp]
      theorem NNReal.tendsto_coe {α : Type u_1} {f : Filter α} {m : αNNReal} {x : NNReal} :
      Filter.Tendsto (fun (a : α) => (m a)) f (nhds x) Filter.Tendsto m f (nhds x)
      theorem NNReal.tendsto_coe' {α : Type u_1} {f : Filter α} [f.NeBot] {m : αNNReal} {x : } :
      Filter.Tendsto (fun (a : α) => (m a)) f (nhds x) ∃ (hx : 0 x), Filter.Tendsto m f (nhds x, hx)
      @[simp]
      theorem NNReal.map_coe_atTop :
      Filter.map NNReal.toReal Filter.atTop = Filter.atTop
      @[simp]
      theorem NNReal.comap_coe_atTop :
      Filter.comap NNReal.toReal Filter.atTop = Filter.atTop
      @[simp]
      theorem NNReal.tendsto_coe_atTop {α : Type u_1} {f : Filter α} {m : αNNReal} :
      Filter.Tendsto (fun (a : α) => (m a)) f Filter.atTop Filter.Tendsto m f Filter.atTop
      theorem tendsto_real_toNNReal {α : Type u_1} {f : Filter α} {m : α} {x : } (h : Filter.Tendsto m f (nhds x)) :
      Filter.Tendsto (fun (a : α) => (m a).toNNReal) f (nhds x.toNNReal)
      @[simp]
      theorem Real.map_toNNReal_atTop :
      Filter.map Real.toNNReal Filter.atTop = Filter.atTop
      @[simp]
      theorem Real.comap_toNNReal_atTop :
      Filter.comap Real.toNNReal Filter.atTop = Filter.atTop
      @[simp]
      theorem Real.tendsto_toNNReal_atTop_iff {α : Type u_1} {l : Filter α} {f : α} :
      Filter.Tendsto (fun (x : α) => (f x).toNNReal) l Filter.atTop Filter.Tendsto f l Filter.atTop
      theorem NNReal.nhds_zero :
      nhds 0 = ⨅ (a : NNReal), ⨅ (_ : a 0), Filter.principal (Set.Iio a)
      theorem NNReal.nhds_zero_basis :
      (nhds 0).HasBasis (fun (a : NNReal) => 0 < a) fun (a : NNReal) => Set.Iio a
      theorem NNReal.hasSum_coe {α : Type u_1} {f : αNNReal} {r : NNReal} :
      HasSum (fun (a : α) => (f a)) r HasSum f r
      theorem HasSum.toNNReal {α : Type u_1} {f : α} {y : } (hf₀ : ∀ (n : α), 0 f n) (hy : HasSum f y) :
      HasSum (fun (x : α) => (f x).toNNReal) y.toNNReal
      theorem NNReal.hasSum_real_toNNReal_of_nonneg {α : Type u_1} {f : α} (hf_nonneg : ∀ (n : α), 0 f n) (hf : Summable f) :
      HasSum (fun (n : α) => (f n).toNNReal) (∑' (n : α), f n).toNNReal
      theorem NNReal.summable_coe {α : Type u_1} {f : αNNReal} :
      (Summable fun (a : α) => (f a)) Summable f
      theorem NNReal.summable_mk {α : Type u_1} {f : α} (hf : ∀ (n : α), 0 f n) :
      (Summable fun (n : α) => f n, ) Summable f
      theorem NNReal.coe_tsum {α : Type u_1} {f : αNNReal} :
      (∑' (a : α), f a) = ∑' (a : α), (f a)
      theorem NNReal.coe_tsum_of_nonneg {α : Type u_1} {f : α} (hf₁ : ∀ (n : α), 0 f n) :
      ∑' (n : α), f n, = ∑' (n : α), f n,
      theorem NNReal.tsum_mul_left {α : Type u_1} (a : NNReal) (f : αNNReal) :
      ∑' (x : α), a * f x = a * ∑' (x : α), f x
      theorem NNReal.tsum_mul_right {α : Type u_1} (f : αNNReal) (a : NNReal) :
      ∑' (x : α), f x * a = (∑' (x : α), f x) * a
      theorem NNReal.summable_comp_injective {α : Type u_1} {β : Type u_2} {f : αNNReal} (hf : Summable f) {i : βα} (hi : Function.Injective i) :
      theorem NNReal.summable_nat_add (f : NNReal) (hf : Summable f) (k : ) :
      Summable fun (i : ) => f (i + k)
      theorem NNReal.summable_nat_add_iff {f : NNReal} (k : ) :
      (Summable fun (i : ) => f (i + k)) Summable f
      theorem NNReal.hasSum_nat_add_iff {f : NNReal} (k : ) {a : NNReal} :
      HasSum (fun (n : ) => f (n + k)) a HasSum f (a + iFinset.range k, f i)
      theorem NNReal.sum_add_tsum_nat_add {f : NNReal} (k : ) (hf : Summable f) :
      ∑' (i : ), f i = iFinset.range k, f i + ∑' (i : ), f (i + k)
      theorem NNReal.iInf_real_pos_eq_iInf_nnreal_pos {α : Type u_1} [CompleteLattice α] {f : α} :
      ⨅ (n : ), ⨅ (_ : 0 < n), f n = ⨅ (n : NNReal), ⨅ (_ : 0 < n), f n
      theorem NNReal.tendsto_cofinite_zero_of_summable {α : Type u_1} {f : αNNReal} (hf : Summable f) :
      Filter.Tendsto f Filter.cofinite (nhds 0)
      theorem NNReal.tendsto_atTop_zero_of_summable {f : NNReal} (hf : Summable f) :
      Filter.Tendsto f Filter.atTop (nhds 0)
      theorem NNReal.tendsto_tsum_compl_atTop_zero {α : Type u_1} (f : αNNReal) :
      Filter.Tendsto (fun (s : Finset α) => ∑' (b : { x : α // xs }), f b) Filter.atTop (nhds 0)

      The sum over the complement of a finset tends to 0 when the finset grows to cover the whole space. This does not need a summability assumption, as otherwise all sums are zero.

      def NNReal.powOrderIso (n : ) (hn : n 0) :

      x ↦ x ^ n as an order isomorphism of ℝ≥0.

      Equations
      Instances For
        theorem Real.tendsto_of_bddAbove_monotone {f : } (h_bdd : BddAbove (Set.range f)) (h_mon : Monotone f) :
        ∃ (r : ), Filter.Tendsto f Filter.atTop (nhds r)

        A monotone, bounded above sequence f : ℕ → ℝ has a finite limit.

        theorem Real.tendsto_of_bddBelow_antitone {f : } (h_bdd : BddBelow (Set.range f)) (h_ant : Antitone f) :
        ∃ (r : ), Filter.Tendsto f Filter.atTop (nhds r)

        An antitone, bounded below sequence f : ℕ → ℝ has a finite limit.

        theorem NNReal.tendsto_of_antitone {f : NNReal} (h_ant : Antitone f) :
        ∃ (r : NNReal), Filter.Tendsto f Filter.atTop (nhds r)

        An antitone sequence f : ℕ → ℝ≥0 has a finite limit.