# Documentation

Mathlib.Topology.Instances.NNReal

# Topology on ℝ≥0#

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

## Main definitions #

Instances for the following typeclasses are defined:

• TopologicalSpace ℝ≥0
• TopologicalSemiring ℝ≥0
• TopologicalSpace.SecondCountableTopology ℝ≥0
• OrderTopology ℝ≥0
• ContinuousSub ℝ≥0
• HasContinuousInv₀ ℝ≥0 (continuity of x⁻¹ away from 0)
• ContinuousSMul ℝ≥0 α (whenever α has a continuous MulAction ℝ α)

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 ℝ. For example

• tendsto_coe {f : Filter α} {m : α → ℝ≥0} {x : ℝ≥0} : Filter.Tendsto (fun a, (m a : ℝ)) f (𝓝 (x : ℝ)) ↔ Filter.Tendsto m f (𝓝 x)

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

• coe_tsum {f : α → ℝ≥0} : ((∑'a, f a) : ℝ) = (∑'a, (f a : ℝ))

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

@[simp]

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

Instances For
instance NNReal.ContinuousMap.canLift {X : Type u_2} [] :
CanLift () fun f => ∀ (x : X), 0 f x
@[simp]
theorem NNReal.tendsto_coe {α : Type u_1} {f : } {m : αNNReal} {x : NNReal} :
Filter.Tendsto (fun a => ↑(m a)) f (nhds x) Filter.Tendsto m f (nhds x)
theorem NNReal.tendsto_coe' {α : Type u_1} {f : } [] {m : αNNReal} {x : } :
Filter.Tendsto (fun a => ↑(m a)) f (nhds x) hx, Filter.Tendsto m f (nhds { val := x, property := hx })
@[simp]
theorem NNReal.map_coe_atTop :
Filter.map NNReal.toReal Filter.atTop = Filter.atTop
theorem NNReal.comap_coe_atTop :
Filter.comap NNReal.toReal Filter.atTop = Filter.atTop
@[simp]
theorem NNReal.tendsto_coe_atTop {α : Type u_1} {f : } {m : αNNReal} :
Filter.Tendsto (fun a => ↑(m a)) f Filter.atTop Filter.Tendsto m f Filter.atTop
theorem tendsto_real_toNNReal {α : Type u_1} {f : } {m : α} {x : } (h : Filter.Tendsto m f (nhds x)) :
Filter.Tendsto (fun a => Real.toNNReal (m a)) f ()
theorem NNReal.nhds_zero :
nhds 0 = ⨅ (a : NNReal) (_ : a 0),
theorem NNReal.nhds_zero_basis :
Filter.HasBasis (nhds 0) (fun a => 0 < a) fun a =>
theorem NNReal.hasSum_coe {α : Type u_1} {f : αNNReal} {r : NNReal} :
HasSum (fun a => ↑(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 => Real.toNNReal (f x)) ()
theorem NNReal.hasSum_real_toNNReal_of_nonneg {α : Type u_1} {f : α} (hf_nonneg : ∀ (n : α), 0 f n) (hf : ) :
HasSum (fun n => Real.toNNReal (f n)) (Real.toNNReal (∑' (n : α), f n))
theorem NNReal.summable_coe {α : Type u_1} {f : αNNReal} :
(Summable fun a => ↑(f a))
theorem NNReal.summable_mk {α : Type u_1} {f : α} (hf : ∀ (n : α), 0 f n) :
(Summable fun n => { val := f n, property := hf n })
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) :
{ val := ∑' (n : α), f n, property := (_ : 0 ∑' (i : α), f i) } = ∑' (n : α), { val := f n, property := hf₁ 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 : ) {i : βα} (hi : ) :
theorem NNReal.summable_nat_add (f : ) (hf : ) (k : ) :
Summable fun i => f (i + k)
theorem NNReal.summable_nat_add_iff {f : } (k : ) :
(Summable fun i => f (i + k))
theorem NNReal.hasSum_nat_add_iff {f : } (k : ) {a : NNReal} :
HasSum (fun n => f (n + k)) a HasSum f (a + Finset.sum () fun i => f i)
theorem NNReal.sum_add_tsum_nat_add {f : } (k : ) (hf : ) :
∑' (i : ), f i = (Finset.sum () fun i => f i) + ∑' (i : ), f (i + k)
theorem NNReal.iInf_real_pos_eq_iInf_nnreal_pos {α : Type u_1} [] {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 : ) :
Filter.Tendsto f Filter.cofinite (nhds 0)
theorem NNReal.tendsto_atTop_zero_of_summable {f : } (hf : ) :
Filter.Tendsto f Filter.atTop (nhds 0)
theorem NNReal.tendsto_tsum_compl_atTop_zero {α : Type u_1} (f : αNNReal) :
Filter.Tendsto (fun s => ∑' (b : { x // ¬x s }), 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.

Instances For