Infinite sum over a topological monoid #
This sum is known as unconditionally convergent, as it sums to the same value under all possible permutations. For Euclidean spaces (finite dimensional Banach spaces) this is equivalent to absolute convergence.
Note: There are summable sequences which are not unconditionally convergent! The other way holds
generally, see has_sum.tendsto_sum_nat.
References #
- Bourbaki: General Topology (1995), Chapter 3 §5 (Infinite sums in commutative groups)
def
has_sum
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
(f : β → α)
(a : α) :
Prop
Infinite sum on a topological monoid
The at_top filter on finset β is the limit of all finite sets towards the entire type. So we sum
up bigger and bigger sets. This sum operation is invariant under reordering. In particular,
the function ℕ → ℝ sending n to (-1)^n / (n+1) does not have a
sum for this definition, but a series which is absolutely convergent will have the correct sum.
This is based on Mario Carneiro's
infinite sum df-tsms in Metamath.
For the definition or many statements, α does not need to be a topological monoid. We only add
this assumption later, for the lemmas where it is relevant.
Equations
- has_sum f a = filter.tendsto (λ (s : finset β), ∑ (b : β) in s, f b) filter.at_top (𝓝 a)
def
summable
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
(f : β → α) :
Prop
summable f means that f has some (infinite) sum. Use tsum to get the value.
noncomputable
def
tsum
{α : Type u_1}
[add_comm_monoid α]
[topological_space α]
{β : Type u_2}
(f : β → α) :
α
∑' i, f i is the sum of f it exists, or 0 otherwise
theorem
summable.has_sum
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
(ha : summable f) :
has_sum f (∑' (b : β), f b)
theorem
has_sum.summable
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
(h : has_sum f a) :
summable f
theorem
has_sum_zero
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α] :
has_sum (λ (b : β), 0) 0
Constant zero function has sum 0
theorem
has_sum_empty
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
[is_empty β] :
has_sum f 0
theorem
summable_zero
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α] :
summable (λ (b : β), 0)
theorem
summable_empty
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
[is_empty β] :
summable f
theorem
tsum_eq_zero_of_not_summable
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
(h : ¬summable f) :
∑' (b : β), f b = 0
theorem
summable_congr
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f g : β → α}
(hfg : ∀ (b : β), f b = g b) :
theorem
summable.congr
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f g : β → α}
(hf : summable f)
(hfg : ∀ (b : β), f b = g b) :
summable g
theorem
has_sum.has_sum_of_sum_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{g : γ → α}
(h_eq : ∀ (u : finset γ), ∃ (v : finset β), ∀ (v' : finset β), v ⊆ v' → (∃ (u' : finset γ), u ⊆ u' ∧ ∑ (x : γ) in u', g x = ∑ (b : β) in v', f b))
(hf : has_sum g a) :
has_sum f a
theorem
has_sum_iff_has_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{g : γ → α}
(h₁ : ∀ (u : finset γ), ∃ (v : finset β), ∀ (v' : finset β), v ⊆ v' → (∃ (u' : finset γ), u ⊆ u' ∧ ∑ (x : γ) in u', g x = ∑ (b : β) in v', f b))
(h₂ : ∀ (v : finset β), ∃ (u : finset γ), ∀ (u' : finset γ), u ⊆ u' → (∃ (v' : finset β), v ⊆ v' ∧ ∑ (b : β) in v', f b = ∑ (x : γ) in u', g x)) :
theorem
function.injective.has_sum_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{g : γ → β}
(hg : function.injective g)
(hf : ∀ (x : β), x ∉ set.range g → f x = 0) :
theorem
function.injective.summable_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{g : γ → β}
(hg : function.injective g)
(hf : ∀ (x : β), x ∉ set.range g → f x = 0) :
theorem
has_sum_subtype_iff_of_support_subset
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{s : set β}
(hf : function.support f ⊆ s) :
theorem
has_sum_subtype_iff_indicator
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{s : set β} :
@[simp]
theorem
has_sum_subtype_support
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α} :
theorem
has_sum_fintype
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[fintype β]
(f : β → α) :
has_sum f (∑ (b : β), f b)
@[protected]
theorem
finset.has_sum
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
(s : finset β)
(f : β → α) :
@[protected]
theorem
finset.summable
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
(s : finset β)
(f : β → α) :
@[protected]
theorem
set.finite.summable
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{s : set β}
(hs : s.finite)
(f : β → α) :
theorem
has_sum_sum_of_ne_finset_zero
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{s : finset β}
(hf : ∀ (b : β), b ∉ s → f b = 0) :
has_sum f (∑ (b : β) in s, f b)
If a function f vanishes outside of a finite set s, then it has_sum ∑ b in s, f b.
theorem
summable_of_ne_finset_zero
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{s : finset β}
(hf : ∀ (b : β), b ∉ s → f b = 0) :
summable f
theorem
has_sum_single
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
(b : β)
(hf : ∀ (b' : β), b' ≠ b → f b' = 0) :
has_sum f (f b)
theorem
has_sum_ite_eq
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
(b : β)
[decidable_pred (λ (_x : β), _x = b)]
(a : α) :
theorem
equiv.has_sum_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
(e : γ ≃ β) :
theorem
function.injective.has_sum_range_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{g : γ → β}
(hg : function.injective g) :
theorem
equiv.summable_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
(e : γ ≃ β) :
theorem
summable.prod_symm
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β × γ → α}
(hf : summable f) :
theorem
equiv.has_sum_iff_of_support
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{g : γ → α}
(e : ↥(function.support f) ≃ ↥(function.support g))
(he : ∀ (x : ↥(function.support f)), g ↑(⇑e x) = f ↑x) :
theorem
has_sum_iff_has_sum_of_ne_zero_bij
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
{g : γ → α}
(i : ↥(function.support g) → β)
(hi : ∀ ⦃x y : ↥(function.support g)⦄, i x = i y → ↑x = ↑y)
(hf : function.support f ⊆ set.range i)
(hfg : ∀ (x : ↥(function.support g)), f (i x) = g ↑x) :
theorem
equiv.summable_iff_of_support
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{g : γ → α}
(e : ↥(function.support f) ≃ ↥(function.support g))
(he : ∀ (x : ↥(function.support f)), g ↑(⇑e x) = f ↑x) :
@[protected]
theorem
has_sum.map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a : α}
[add_comm_monoid γ]
[topological_space γ]
(hf : has_sum f a)
{G : Type u_4}
[add_monoid_hom_class G α γ]
(g : G)
(hg : continuous ⇑g) :
@[protected]
theorem
summable.map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
[add_comm_monoid γ]
[topological_space γ]
(hf : summable f)
{G : Type u_4}
[add_monoid_hom_class G α γ]
(g : G)
(hg : continuous ⇑g) :
theorem
has_sum.tendsto_sum_nat
{α : Type u_1}
[add_comm_monoid α]
[topological_space α]
{a : α}
{f : ℕ → α}
(h : has_sum f a) :
filter.tendsto (λ (n : ℕ), ∑ (i : ℕ) in finset.range n, f i) filter.at_top (𝓝 a)
If f : ℕ → α has sum a, then the partial sums ∑_{i=0}^{n-1} f i converge to a.
theorem
has_sum.unique
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a₁ a₂ : α}
[t2_space α] :
theorem
summable.has_sum_iff_tendsto_nat
{α : Type u_1}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : ℕ → α}
{a : α}
(hf : summable f) :
has_sum f a ↔ filter.tendsto (λ (n : ℕ), ∑ (i : ℕ) in finset.range n, f i) filter.at_top (𝓝 a)
theorem
function.surjective.summable_iff_of_has_sum_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
{α' : Type u_4}
[add_comm_monoid α']
[topological_space α']
{e : α' → α}
(hes : function.surjective e)
{f : β → α}
{g : γ → α'}
(he : ∀ {a : α'}, has_sum f (e a) ↔ has_sum g a) :
theorem
has_sum.add
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f g : β → α}
{a b : α}
[has_continuous_add α]
(hf : has_sum f a)
(hg : has_sum g b) :
theorem
summable.add
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f g : β → α}
[has_continuous_add α]
(hf : summable f)
(hg : summable g) :
theorem
has_sum_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
{f : γ → β → α}
{a : γ → α}
{s : finset γ} :
theorem
summable_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
{f : γ → β → α}
{s : finset γ}
(hf : ∀ (i : γ), i ∈ s → summable (f i)) :
summable (λ (b : β), ∑ (i : γ) in s, f i b)
theorem
has_sum.add_disjoint
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a b : α}
[has_continuous_add α]
{s t : set β}
(hs : disjoint s t)
(ha : has_sum (f ∘ coe) a)
(hb : has_sum (f ∘ coe) b) :
theorem
has_sum.add_is_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a b : α}
[has_continuous_add α]
{s t : set β}
(hs : is_compl s t)
(ha : has_sum (f ∘ coe) a)
(hb : has_sum (f ∘ coe) b) :
theorem
has_sum.add_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a b : α}
[has_continuous_add α]
{s : set β}
(ha : has_sum (f ∘ coe) a)
(hb : has_sum (f ∘ coe) b) :
theorem
summable.add_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
[has_continuous_add α]
{s : set β}
(hs : summable (f ∘ coe))
(hsc : summable (f ∘ coe)) :
summable f
theorem
has_sum.compl_add
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
{a b : α}
[has_continuous_add α]
{s : set β}
(ha : has_sum (f ∘ coe) a)
(hb : has_sum (f ∘ coe) b) :
theorem
has_sum.even_add_odd
{α : Type u_1}
[add_comm_monoid α]
[topological_space α]
{a b : α}
[has_continuous_add α]
{f : ℕ → α}
(he : has_sum (λ (k : ℕ), f (2 * k)) a)
(ho : has_sum (λ (k : ℕ), f (2 * k + 1)) b) :
theorem
summable.compl_add
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f : β → α}
[has_continuous_add α]
{s : set β}
(hs : summable (f ∘ coe))
(hsc : summable (f ∘ coe)) :
summable f
theorem
summable.even_add_odd
{α : Type u_1}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
{f : ℕ → α}
(he : summable (λ (k : ℕ), f (2 * k)))
(ho : summable (λ (k : ℕ), f (2 * k + 1))) :
summable f
theorem
has_sum.sigma
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
[regular_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
{g : β → α}
{a : α}
(ha : has_sum f a)
(hf : ∀ (b : β), has_sum (λ (c : γ b), f ⟨b, c⟩) (g b)) :
has_sum g a
theorem
has_sum.prod_fiberwise
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
[regular_space α]
{f : β × γ → α}
{g : β → α}
{a : α}
(ha : has_sum f a)
(hf : ∀ (b : β), has_sum (λ (c : γ), f (b, c)) (g b)) :
has_sum g a
If a series f on β × γ has sum a and for each b the restriction of f to {b} × γ
has sum g b, then the series g has sum a.
theorem
summable.sigma'
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
[regular_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
(ha : summable f)
(hf : ∀ (b : β), summable (λ (c : γ b), f ⟨b, c⟩)) :
theorem
has_sum.sigma_of_has_sum
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[has_continuous_add α]
[regular_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
{g : β → α}
{a : α}
(ha : has_sum g a)
(hf : ∀ (b : β), has_sum (λ (c : γ b), f ⟨b, c⟩) (g b))
(hf' : summable f) :
has_sum f a
theorem
tsum_congr_subtype
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
(f : β → α)
{s t : set β}
(h : s = t) :
theorem
has_sum.tsum_eq
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
{a : α}
(ha : has_sum f a) :
∑' (b : β), f b = a
theorem
summable.has_sum_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
{a : α}
(h : summable f) :
@[simp]
theorem
tsum_zero
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α] :
∑' (b : β), 0 = 0
@[simp]
theorem
tsum_empty
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
[is_empty β] :
∑' (b : β), f b = 0
theorem
tsum_eq_sum
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
{s : finset β}
(hf : ∀ (b : β), b ∉ s → f b = 0) :
∑' (b : β), f b = ∑ (b : β) in s, f b
theorem
tsum_congr
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
{f g : β → α}
(hfg : ∀ (b : β), f b = g b) :
∑' (b : β), f b = ∑' (b : β), g b
theorem
tsum_fintype
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[fintype β]
(f : β → α) :
∑' (b : β), f b = ∑ (b : β), f b
@[simp]
theorem
finset.tsum_subtype
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(s : finset β)
(f : β → α) :
@[simp]
theorem
finset.tsum_subtype'
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(s : finset β)
(f : β → α) :
theorem
tsum_eq_single
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
(b : β)
(hf : ∀ (b' : β), b' ≠ b → f b' = 0) :
∑' (b : β), f b = f b
@[simp]
theorem
tsum_ite_eq
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(b : β)
[decidable_pred (λ (_x : β), _x = b)]
(a : α) :
theorem
tsum_dite_right
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(P : Prop)
[decidable P]
(x : β → ¬P → α) :
theorem
tsum_dite_left
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(P : Prop)
[decidable P]
(x : β → P → α) :
theorem
function.surjective.tsum_eq_tsum_of_has_sum_iff_has_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{α' : Type u_4}
[add_comm_monoid α']
[topological_space α']
{e : α' → α}
(hes : function.surjective e)
(h0 : e 0 = 0)
{f : β → α}
{g : γ → α'}
(h : ∀ {a : α'}, has_sum f (e a) ↔ has_sum g a) :
∑' (b : β), f b = e (∑' (c : γ), g c)
theorem
tsum_eq_tsum_of_has_sum_iff_has_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
{g : γ → α}
(h : ∀ {a : α}, has_sum f a ↔ has_sum g a) :
∑' (b : β), f b = ∑' (c : γ), g c
theorem
equiv.tsum_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(j : γ ≃ β)
(f : β → α) :
theorem
equiv.tsum_eq_tsum_of_support
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
{g : γ → α}
(e : ↥(function.support f) ≃ ↥(function.support g))
(he : ∀ (x : ↥(function.support f)), g ↑(⇑e x) = f ↑x) :
∑' (x : β), f x = ∑' (y : γ), g y
theorem
tsum_eq_tsum_of_ne_zero_bij
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
{g : γ → α}
(i : ↥(function.support g) → β)
(hi : ∀ ⦃x y : ↥(function.support g)⦄, i x = i y → ↑x = ↑y)
(hf : function.support f ⊆ set.range i)
(hfg : ∀ (x : ↥(function.support g)), f (i x) = g ↑x) :
∑' (x : β), f x = ∑' (y : γ), g y
theorem
tsum_subtype
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
(s : set β)
(f : β → α) :
theorem
tsum_add
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f g : β → α}
[has_continuous_add α]
(hf : summable f)
(hg : summable g) :
theorem
tsum_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[has_continuous_add α]
{f : γ → β → α}
{s : finset γ}
(hf : ∀ (i : γ), i ∈ s → summable (f i)) :
∑' (b : β), ∑ (i : γ) in s, f i b = ∑ (i : γ) in s, ∑' (b : β), f i b
theorem
tsum_sigma'
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[has_continuous_add α]
[regular_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
(h₁ : ∀ (b : β), summable (λ (c : γ b), f ⟨b, c⟩))
(h₂ : summable f) :
theorem
tsum_prod'
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[has_continuous_add α]
[regular_space α]
{f : β × γ → α}
(h : summable f)
(h₁ : ∀ (b : β), summable (λ (c : γ), f (b, c))) :
theorem
tsum_comm'
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[has_continuous_add α]
[regular_space α]
{f : β → γ → α}
(h : summable (function.uncurry f))
(h₁ : ∀ (b : β), summable (f b))
(h₂ : ∀ (c : γ), summable (λ (b : β), f b c)) :
∑' (c : γ) (b : β), f b c = ∑' (b : β) (c : γ), f b c
theorem
tsum_supr_decode₂
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[encodable γ]
[complete_lattice β]
(m : β → α)
(m0 : m ⊥ = 0)
(s : γ → β) :
∑' (i : ℕ), m (⨆ (b : γ) (H : b ∈ encodable.decode₂ γ i), s b) = ∑' (b : γ), m (s b)
You can compute a sum over an encodably type by summing over the natural numbers and taking a supremum. This is useful for outer measures.
theorem
tsum_Union_decode₂
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[encodable γ]
(m : set β → α)
(m0 : m ∅ = 0)
(s : γ → set β) :
∑' (i : ℕ), m (⋃ (b : γ) (H : b ∈ encodable.decode₂ γ i), s b) = ∑' (b : γ), m (s b)
tsum_supr_decode₂ specialized to the complete lattice of sets.
Some properties about measure-like functions.
These could also be functions defined on complete sublattices of sets, with the property
that they are countably sub-additive.
R will probably be instantiated with (≤) in all applications.
theorem
rel_supr_tsum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[encodable γ]
[complete_lattice β]
(m : β → α)
(m0 : m ⊥ = 0)
(R : α → α → Prop)
(m_supr : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i)))
(s : γ → β) :
R (m (⨆ (b : γ), s b)) (∑' (b : γ), m (s b))
If a function is countably sub-additive then it is sub-additive on encodable types
theorem
rel_supr_sum
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[complete_lattice β]
(m : β → α)
(m0 : m ⊥ = 0)
(R : α → α → Prop)
(m_supr : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i)))
(s : δ → β)
(t : finset δ) :
R (m (⨆ (d : δ) (H : d ∈ t), s d)) (∑ (d : δ) in t, m (s d))
If a function is countably sub-additive then it is sub-additive on finite sets
theorem
rel_sup_add
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
[complete_lattice β]
(m : β → α)
(m0 : m ⊥ = 0)
(R : α → α → Prop)
(m_supr : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i)))
(s₁ s₂ : β) :
If a function is countably sub-additive then it is binary sub-additive
theorem
tsum_add_tsum_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_monoid α]
[topological_space α]
[t2_space α]
{f : β → α}
[has_continuous_add α]
{s : set β}
(hs : summable (f ∘ coe))
(hsc : summable (f ∘ coe)) :
theorem
has_sum.prod_mk
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_monoid α]
[topological_space α]
[add_comm_monoid γ]
[topological_space γ]
{f : β → α}
{g : β → γ}
{a : α}
{b : γ}
(hf : has_sum f a)
(hg : has_sum g b) :
has_sum (λ (x : β), (f x, g x)) (a, b)
theorem
pi.has_sum
{α : Type u_1}
{ι : Type u_5}
{π : α → Type u_6}
[Π (x : α), add_comm_monoid (π x)]
[Π (x : α), topological_space (π x)]
{f : ι → Π (x : α), π x}
{g : Π (x : α), π x} :
theorem
pi.summable
{α : Type u_1}
{ι : Type u_5}
{π : α → Type u_6}
[Π (x : α), add_comm_monoid (π x)]
[Π (x : α), topological_space (π x)]
{f : ι → Π (x : α), π x} :
theorem
tsum_apply
{α : Type u_1}
{ι : Type u_5}
{π : α → Type u_6}
[Π (x : α), add_comm_monoid (π x)]
[Π (x : α), topological_space (π x)]
[∀ (x : α), t2_space (π x)]
{f : ι → Π (x : α), π x}
{x : α}
(hf : summable f) :
(∑' (i : ι), f i) x = ∑' (i : ι), f i x
theorem
has_sum.neg
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a : α}
(h : has_sum f a) :
theorem
summable.neg
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
(hf : summable f) :
theorem
summable.of_neg
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
(hf : summable (λ (b : β), -f b)) :
summable f
theorem
summable_neg_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α} :
theorem
has_sum.sub
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f g : β → α}
{a₁ a₂ : α}
(hf : has_sum f a₁)
(hg : has_sum g a₂) :
theorem
summable.sub
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f g : β → α}
(hf : summable f)
(hg : summable g) :
theorem
summable.trans_sub
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f g : β → α}
(hg : summable g)
(hfg : summable (λ (b : β), f b - g b)) :
summable f
theorem
summable_iff_of_summable_sub
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f g : β → α}
(hfg : summable (λ (b : β), f b - g b)) :
theorem
has_sum.update
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a₁ : α}
(hf : has_sum f a₁)
(b : β)
[decidable_eq β]
(a : α) :
has_sum (function.update f b a) (a - f b + a₁)
theorem
summable.update
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
(hf : summable f)
(b : β)
[decidable_eq β]
(a : α) :
summable (function.update f b a)
theorem
has_sum.has_sum_compl_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a₁ a₂ : α}
{s : set β}
(hf : has_sum (f ∘ coe) a₁) :
theorem
has_sum.has_sum_iff_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a₁ a₂ : α}
{s : set β}
(hf : has_sum (f ∘ coe) a₁) :
theorem
summable.summable_compl_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{s : set β}
(hf : summable (f ∘ coe)) :
@[protected]
theorem
finset.has_sum_compl_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a : α}
(s : finset β) :
@[protected]
theorem
finset.has_sum_iff_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a : α}
(s : finset β) :
@[protected]
theorem
finset.summable_compl_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
(s : finset β) :
theorem
set.finite.summable_compl_iff
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{s : set β}
(hs : s.finite) :
theorem
has_sum_ite_eq_extract
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
{a : α}
[decidable_eq β]
(hf : has_sum f a)
(b : β) :
theorem
tsum_neg
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
[t2_space α]
(hf : summable f) :
theorem
tsum_sub
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f g : β → α}
[t2_space α]
(hf : summable f)
(hg : summable g) :
theorem
sum_add_tsum_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
[t2_space α]
{s : finset β}
(hf : summable f) :
theorem
tsum_ite_eq_extract
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : β → α}
[t2_space α]
[decidable_eq β]
(hf : summable f)
(b : β) :
Let f : β → α be a sequence with summable series and let b ∈ β be an index.
Lemma tsum_ite_eq_extract writes Σ f n as the sum of f b plus the series of the
remaining terms.
Sums on subtypes #
If s is a finset of α, we show that the summability of f in the whole space and on the subtype
univ - s are equivalent, and relate their sums. For a function defined on ℕ, we deduce the
formula (∑ i in range k, f i) + (∑' i, f (i + k)) = (∑' i, f i), in sum_add_tsum_nat_add.
theorem
has_sum_nat_add_iff
{α : Type u_1}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : ℕ → α}
(k : ℕ)
{a : α} :
theorem
summable_nat_add_iff
{α : Type u_1}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : ℕ → α}
(k : ℕ) :
theorem
has_sum_nat_add_iff'
{α : Type u_1}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
{f : ℕ → α}
(k : ℕ)
{a : α} :
theorem
sum_add_tsum_nat_add
{α : Type u_1}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
[t2_space α]
{f : ℕ → α}
(k : ℕ)
(h : summable f) :
theorem
tsum_eq_zero_add
{α : Type u_1}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
[t2_space α]
{f : ℕ → α}
(hf : summable f) :
theorem
tendsto_sum_nat_add
{α : Type u_1}
[add_comm_group α]
[topological_space α]
[topological_add_group α]
[t2_space α]
(f : ℕ → α) :
filter.tendsto (λ (i : ℕ), ∑' (k : ℕ), f (k + i)) filter.at_top (𝓝 0)
For f : ℕ → α, then ∑' k, f (k + i) tends to zero. This does not require a summability
assumption on f, as otherwise all sums are zero.
theorem
has_sum.mul_left
{α : Type u_1}
{β : Type u_2}
[non_unital_non_assoc_semiring α]
[topological_space α]
[topological_semiring α]
{f : β → α}
{a₁ : α}
(a₂ : α)
(h : has_sum f a₁) :
theorem
has_sum.mul_right
{α : Type u_1}
{β : Type u_2}
[non_unital_non_assoc_semiring α]
[topological_space α]
[topological_semiring α]
{f : β → α}
{a₁ : α}
(a₂ : α)
(hf : has_sum f a₁) :
theorem
summable.mul_left
{α : Type u_1}
{β : Type u_2}
[non_unital_non_assoc_semiring α]
[topological_space α]
[topological_semiring α]
{f : β → α}
(a : α)
(hf : summable f) :
theorem
summable.mul_right
{α : Type u_1}
{β : Type u_2}
[non_unital_non_assoc_semiring α]
[topological_space α]
[topological_semiring α]
{f : β → α}
(a : α)
(hf : summable f) :
theorem
summable.tsum_mul_left
{α : Type u_1}
{β : Type u_2}
[non_unital_non_assoc_semiring α]
[topological_space α]
[topological_semiring α]
{f : β → α}
[t2_space α]
(a : α)
(hf : summable f) :
theorem
summable.tsum_mul_right
{α : Type u_1}
{β : Type u_2}
[non_unital_non_assoc_semiring α]
[topological_space α]
[topological_semiring α]
{f : β → α}
[t2_space α]
(a : α)
(hf : summable f) :
theorem
has_sum.const_smul
{α : Type u_1}
{β : Type u_2}
{R : Type u_5}
[monoid R]
[topological_space α]
[add_comm_monoid α]
[distrib_mul_action R α]
[has_continuous_const_smul R α]
{f : β → α}
{a : α}
{r : R}
(hf : has_sum f a) :
theorem
summable.const_smul
{α : Type u_1}
{β : Type u_2}
{R : Type u_5}
[monoid R]
[topological_space α]
[add_comm_monoid α]
[distrib_mul_action R α]
[has_continuous_const_smul R α]
{f : β → α}
{r : R}
(hf : summable f) :
theorem
tsum_const_smul
{α : Type u_1}
{β : Type u_2}
{R : Type u_5}
[monoid R]
[topological_space α]
[add_comm_monoid α]
[distrib_mul_action R α]
[has_continuous_const_smul R α]
{f : β → α}
[t2_space α]
{r : R}
(hf : summable f) :
theorem
has_sum.smul_const
{α : Type u_1}
{β : Type u_2}
{R : Type u_5}
[semiring R]
[topological_space R]
[topological_space α]
[add_comm_monoid α]
[module R α]
[has_continuous_smul R α]
{f : β → R}
{a : α}
{r : R}
(hf : has_sum f r) :
theorem
summable.smul_const
{α : Type u_1}
{β : Type u_2}
{R : Type u_5}
[semiring R]
[topological_space R]
[topological_space α]
[add_comm_monoid α]
[module R α]
[has_continuous_smul R α]
{f : β → R}
{a : α}
(hf : summable f) :
theorem
tsum_smul_const
{α : Type u_1}
{β : Type u_2}
{R : Type u_5}
[semiring R]
[topological_space R]
[topological_space α]
[add_comm_monoid α]
[module R α]
[has_continuous_smul R α]
{f : β → R}
[t2_space α]
{a : α}
(hf : summable f) :
theorem
has_sum.div_const
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a : α}
(h : has_sum f a)
(b : α) :
theorem
summable.div_const
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
(h : summable f)
(b : α) :
theorem
has_sum_mul_left_iff
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a₁ a₂ : α}
(h : a₂ ≠ 0) :
theorem
has_sum_mul_right_iff
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a₁ a₂ : α}
(h : a₂ ≠ 0) :
theorem
summable_mul_left_iff
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a : α}
(h : a ≠ 0) :
theorem
summable_mul_right_iff
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a : α}
(h : a ≠ 0) :
theorem
tsum_mul_left
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a : α}
[t2_space α] :
theorem
tsum_mul_right
{α : Type u_1}
{β : Type u_2}
[division_ring α]
[topological_space α]
[topological_ring α]
{f : β → α}
{a : α}
[t2_space α] :
theorem
has_sum_le
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f g : β → α}
{a₁ a₂ : α}
(h : ∀ (b : β), f b ≤ g b)
(hf : has_sum f a₁)
(hg : has_sum g a₂) :
a₁ ≤ a₂
theorem
has_sum_mono
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f g : β → α}
{a₁ a₂ : α}
(hf : has_sum f a₁)
(hg : has_sum g a₂)
(h : f ≤ g) :
a₁ ≤ a₂
theorem
has_sum_le_of_sum_le
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a a₂ : α}
(hf : has_sum f a)
(h : ∀ (s : finset β), ∑ (b : β) in s, f b ≤ a₂) :
a ≤ a₂
theorem
le_has_sum_of_le_sum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a a₂ : α}
(hf : has_sum f a)
(h : ∀ (s : finset β), a₂ ≤ ∑ (b : β) in s, f b) :
a₂ ≤ a
theorem
has_sum_le_inj
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a₁ a₂ : α}
{g : γ → α}
(i : β → γ)
(hi : function.injective i)
(hs : ∀ (c : γ), c ∉ set.range i → 0 ≤ g c)
(h : ∀ (b : β), f b ≤ g (i b))
(hf : has_sum f a₁)
(hg : has_sum g a₂) :
a₁ ≤ a₂
theorem
tsum_le_tsum_of_inj
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{g : γ → α}
(i : β → γ)
(hi : function.injective i)
(hs : ∀ (c : γ), c ∉ set.range i → 0 ≤ g c)
(h : ∀ (b : β), f b ≤ g (i b))
(hf : summable f)
(hg : summable g) :
theorem
sum_le_has_sum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a : α}
(s : finset β)
(hs : ∀ (b : β), b ∉ s → 0 ≤ f b)
(hf : has_sum f a) :
∑ (b : β) in s, f b ≤ a
theorem
is_lub_has_sum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a : α}
(h : ∀ (b : β), 0 ≤ f b)
(hf : has_sum f a) :
theorem
le_has_sum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a : α}
(hf : has_sum f a)
(b : β)
(hb : ∀ (b' : β), b' ≠ b → 0 ≤ f b') :
f b ≤ a
theorem
sum_le_tsum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
(s : finset β)
(hs : ∀ (b : β), b ∉ s → 0 ≤ f b)
(hf : summable f) :
∑ (b : β) in s, f b ≤ ∑' (b : β), f b
theorem
le_tsum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
(hf : summable f)
(b : β)
(hb : ∀ (b' : β), b' ≠ b → 0 ≤ f b') :
f b ≤ ∑' (b : β), f b
theorem
tsum_le_tsum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f g : β → α}
(h : ∀ (b : β), f b ≤ g b)
(hf : summable f)
(hg : summable g) :
∑' (b : β), f b ≤ ∑' (b : β), g b
theorem
tsum_mono
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f g : β → α}
(hf : summable f)
(hg : summable g)
(h : f ≤ g) :
∑' (n : β), f n ≤ ∑' (n : β), g n
theorem
tsum_le_of_sum_le
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a₂ : α}
(hf : summable f)
(h : ∀ (s : finset β), ∑ (b : β) in s, f b ≤ a₂) :
∑' (b : β), f b ≤ a₂
theorem
tsum_le_of_sum_le'
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a₂ : α}
(ha₂ : 0 ≤ a₂)
(h : ∀ (s : finset β), ∑ (b : β) in s, f b ≤ a₂) :
∑' (b : β), f b ≤ a₂
theorem
has_sum.nonneg
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{g : β → α}
{a : α}
(h : ∀ (b : β), 0 ≤ g b)
(ha : has_sum g a) :
0 ≤ a
theorem
has_sum.nonpos
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{g : β → α}
{a : α}
(h : ∀ (b : β), g b ≤ 0)
(ha : has_sum g a) :
a ≤ 0
theorem
tsum_nonneg
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{g : β → α}
(h : ∀ (b : β), 0 ≤ g b) :
0 ≤ ∑' (b : β), g b
theorem
tsum_nonpos
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
(h : ∀ (b : β), f b ≤ 0) :
∑' (b : β), f b ≤ 0
theorem
has_sum_lt
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
[order_closed_topology α]
{f g : β → α}
{a₁ a₂ : α}
{i : β}
(h : ∀ (b : β), f b ≤ g b)
(hi : f i < g i)
(hf : has_sum f a₁)
(hg : has_sum g a₂) :
a₁ < a₂
theorem
has_sum_strict_mono
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
[order_closed_topology α]
{f g : β → α}
{a₁ a₂ : α}
(hf : has_sum f a₁)
(hg : has_sum g a₂)
(h : f < g) :
a₁ < a₂
theorem
tsum_lt_tsum
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
[order_closed_topology α]
{f g : β → α}
{i : β}
(h : ∀ (b : β), f b ≤ g b)
(hi : f i < g i)
(hf : summable f)
(hg : summable g) :
∑' (n : β), f n < ∑' (n : β), g n
theorem
tsum_strict_mono
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
[order_closed_topology α]
{f g : β → α}
(hf : summable f)
(hg : summable g)
(h : f < g) :
∑' (n : β), f n < ∑' (n : β), g n
theorem
tsum_pos
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
[order_closed_topology α]
{g : β → α}
(hsum : summable g)
(hg : ∀ (b : β), 0 ≤ g b)
(i : β)
(hi : 0 < g i) :
0 < ∑' (b : β), g b
theorem
has_sum_zero_iff_of_nonneg
{α : Type u_1}
{β : Type u_2}
[ordered_add_comm_group α]
[topological_space α]
[topological_add_group α]
[order_closed_topology α]
{f : β → α}
(hf : ∀ (i : β), 0 ≤ f i) :
theorem
le_has_sum'
{α : Type u_1}
{β : Type u_2}
[canonically_ordered_add_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a : α}
(hf : has_sum f a)
(b : β) :
f b ≤ a
theorem
le_tsum'
{α : Type u_1}
{β : Type u_2}
[canonically_ordered_add_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
(hf : summable f)
(b : β) :
f b ≤ ∑' (b : β), f b
theorem
has_sum_zero_iff
{α : Type u_1}
{β : Type u_2}
[canonically_ordered_add_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α} :
theorem
tsum_eq_zero_iff
{α : Type u_1}
{β : Type u_2}
[canonically_ordered_add_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
(hf : summable f) :
theorem
tsum_ne_zero_iff
{α : Type u_1}
{β : Type u_2}
[canonically_ordered_add_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
(hf : summable f) :
theorem
is_lub_has_sum'
{α : Type u_1}
{β : Type u_2}
[canonically_ordered_add_monoid α]
[topological_space α]
[order_closed_topology α]
{f : β → α}
{a : α}
(hf : has_sum f a) :
theorem
summable_iff_cauchy_seq_finset
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[complete_space α]
{f : β → α} :
summable f ↔ cauchy_seq (λ (s : finset β), ∑ (b : β) in s, f b)
The Cauchy criterion for infinite sums, also known as the Cauchy convergence test
theorem
cauchy_seq_finset_iff_vanishing
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f : β → α} :
theorem
tendsto_tsum_compl_at_top_zero
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[t1_space α]
(f : β → α) :
filter.tendsto (λ (s : finset β), ∑' (b : {x // x ∉ s}), f ↑b) filter.at_top (𝓝 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.
theorem
summable_iff_vanishing
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f : β → α}
[complete_space α] :
theorem
summable.summable_of_eq_zero_or_self
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f g : β → α}
[complete_space α]
(hf : summable f)
(h : ∀ (b : β), g b = 0 ∨ g b = f b) :
summable g
@[protected]
theorem
summable.indicator
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f : β → α}
[complete_space α]
(hf : summable f)
(s : set β) :
theorem
summable.comp_injective
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f : β → α}
[complete_space α]
{i : γ → β}
(hf : summable f)
(hi : function.injective i) :
theorem
summable.subtype
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f : β → α}
[complete_space α]
(hf : summable f)
(s : set β) :
theorem
summable_subtype_and_compl
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
{f : β → α}
[complete_space α]
{s : set β} :
theorem
summable.sigma_factor
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[complete_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
(ha : summable f)
(b : β) :
theorem
summable.sigma
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[complete_space α]
[t1_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
(ha : summable f) :
theorem
summable.prod_factor
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[complete_space α]
{f : β × γ → α}
(h : summable f)
(b : β) :
summable (λ (c : γ), f (b, c))
theorem
tsum_sigma
{α : Type u_1}
{β : Type u_2}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[complete_space α]
[t1_space α]
{γ : β → Type u_3}
{f : (Σ (b : β), γ b) → α}
(ha : summable f) :
theorem
tsum_prod
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[complete_space α]
[t1_space α]
{f : β × γ → α}
(h : summable f) :
theorem
tsum_comm
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_group α]
[uniform_space α]
[uniform_add_group α]
[complete_space α]
[t1_space α]
{f : β → γ → α}
(h : summable (function.uncurry f)) :
∑' (c : γ) (b : β), f b c = ∑' (b : β) (c : γ), f b c
theorem
summable.vanishing
{α : Type u_1}
{G : Type u_5}
[topological_space G]
[add_comm_group G]
[topological_add_group G]
{f : α → G}
(hf : summable f)
⦃e : set G⦄
(he : e ∈ 𝓝 0) :
theorem
summable.tendsto_cofinite_zero
{α : Type u_1}
{G : Type u_5}
[topological_space G]
[add_comm_group G]
[topological_add_group G]
{f : α → G}
(hf : summable f) :
filter.tendsto f filter.cofinite (𝓝 0)
Series divergence test: if f is a convergent series, then f x tends to zero along
cofinite.
theorem
summable.tendsto_at_top_zero
{G : Type u_5}
[topological_space G]
[add_comm_group G]
[topological_add_group G]
{f : ℕ → G}
(hf : summable f) :
filter.tendsto f filter.at_top (𝓝 0)
theorem
summable.tendsto_top_of_pos
{α : Type u_1}
[linear_ordered_field α]
[topological_space α]
[order_topology α]
{f : ℕ → α}
(hf : summable f⁻¹)
(hf' : ∀ (n : ℕ), 0 < f n) :
For infinite sums taking values in a linearly ordered monoid, the existence of a least upper bound for the finite sums is a criterion for summability.
This criterion is useful when applied in a linearly ordered monoid which is also a complete or
conditionally complete linear order, such as ℝ, ℝ≥0, ℝ≥0∞, because it is then easy to check
the existence of a least upper bound.
theorem
has_sum_of_is_lub_of_nonneg
{α : Type u_1}
{β : Type u_2}
[linear_ordered_add_comm_monoid β]
[topological_space β]
[order_topology β]
{f : α → β}
(b : β)
(h : ∀ (b : α), 0 ≤ f b)
(hf : is_lub (set.range (λ (s : finset α), ∑ (a : α) in s, f a)) b) :
has_sum f b
theorem
has_sum_of_is_lub
{α : Type u_1}
{β : Type u_2}
[canonically_linear_ordered_add_monoid β]
[topological_space β]
[order_topology β]
{f : α → β}
(b : β)
(hf : is_lub (set.range (λ (s : finset α), ∑ (a : α) in s, f a)) b) :
has_sum f b
theorem
summable_abs_iff
{α : Type u_1}
{β : Type u_2}
[linear_ordered_add_comm_group β]
[uniform_space β]
[uniform_add_group β]
[complete_space β]
{f : α → β} :
theorem
summable.abs
{α : Type u_1}
{β : Type u_2}
[linear_ordered_add_comm_group β]
[uniform_space β]
[uniform_add_group β]
[complete_space β]
{f : α → β} :
Alias of summable_abs_iff.
theorem
summable.of_abs
{α : Type u_1}
{β : Type u_2}
[linear_ordered_add_comm_group β]
[uniform_space β]
[uniform_add_group β]
[complete_space β]
{f : α → β} :
Alias of summable_abs_iff.
theorem
finite_of_summable_const
{α : Type u_1}
{β : Type u_2}
[linear_ordered_add_comm_group β]
[archimedean β]
[topological_space β]
[order_closed_topology β]
{b : β}
(hb : 0 < b)
(hf : summable (λ (a : α), b)) :
theorem
cauchy_seq_of_edist_le_of_summable
{α : Type u_1}
[pseudo_emetric_space α]
{f : ℕ → α}
(d : ℕ → ℝ≥0)
(hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n))
(hd : summable d) :
If the extended distance between consecutive points of a sequence is estimated
by a summable series of nnreals, then the original sequence is a Cauchy sequence.
theorem
cauchy_seq_of_dist_le_of_summable
{α : Type u_1}
[pseudo_metric_space α]
{f : ℕ → α}
(d : ℕ → ℝ)
(hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n)
(hd : summable d) :
If the distance between consecutive points of a sequence is estimated by a summable series, then the original sequence is a Cauchy sequence.
theorem
cauchy_seq_of_summable_dist
{α : Type u_1}
[pseudo_metric_space α]
{f : ℕ → α}
(h : summable (λ (n : ℕ), dist (f n) (f n.succ))) :
theorem
dist_le_tsum_of_dist_le_of_tendsto
{α : Type u_1}
[pseudo_metric_space α]
{f : ℕ → α}
(d : ℕ → ℝ)
(hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n)
(hd : summable d)
{a : α}
(ha : filter.tendsto f filter.at_top (𝓝 a))
(n : ℕ) :
theorem
dist_le_tsum_of_dist_le_of_tendsto₀
{α : Type u_1}
[pseudo_metric_space α]
{f : ℕ → α}
(d : ℕ → ℝ)
(hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n)
(hd : summable d)
{a : α}
(ha : filter.tendsto f filter.at_top (𝓝 a)) :
theorem
dist_le_tsum_dist_of_tendsto
{α : Type u_1}
[pseudo_metric_space α]
{f : ℕ → α}
(h : summable (λ (n : ℕ), dist (f n) (f n.succ)))
{a : α}
(ha : filter.tendsto f filter.at_top (𝓝 a))
(n : ℕ) :
theorem
dist_le_tsum_dist_of_tendsto₀
{α : Type u_1}
[pseudo_metric_space α]
{f : ℕ → α}
(h : summable (λ (n : ℕ), dist (f n) (f n.succ)))
{a : α}
(ha : filter.tendsto f filter.at_top (𝓝 a)) :
Multipliying two infinite sums #
In this section, we prove various results about (∑' x : β, f x) * (∑' y : γ, g y). Note that we
always assume that the family λ x : β × γ, f x.1 * g x.2 is summable, since there is no way to
deduce this from the summmabilities of f and g in general, but if you are working in a normed
space, you may want to use the analogous lemmas in analysis/normed_space/basic
(e.g tsum_mul_tsum_of_summable_norm).
We first establish results about arbitrary index types, β and γ, and then we specialize to
β = γ = ℕ to prove the Cauchy product formula (see tsum_mul_tsum_eq_tsum_sum_antidiagonal).
Arbitrary index types #
theorem
has_sum.mul_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[topological_space α]
[regular_space α]
[non_unital_non_assoc_semiring α]
[topological_semiring α]
{f : β → α}
{g : γ → α}
{s t u : α}
(hf : has_sum f s)
(hg : has_sum g t)
(hfg : has_sum (λ (x : β × γ), (f x.fst) * g x.snd) u) :
theorem
has_sum.mul
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[topological_space α]
[regular_space α]
[non_unital_non_assoc_semiring α]
[topological_semiring α]
{f : β → α}
{g : γ → α}
{s t : α}
(hf : has_sum f s)
(hg : has_sum g t)
(hfg : summable (λ (x : β × γ), (f x.fst) * g x.snd)) :
theorem
tsum_mul_tsum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[topological_space α]
[regular_space α]
[non_unital_non_assoc_semiring α]
[topological_semiring α]
{f : β → α}
{g : γ → α}
(hf : summable f)
(hg : summable g)
(hfg : summable (λ (x : β × γ), (f x.fst) * g x.snd)) :
Product of two infinites sums indexed by arbitrary types.
See also tsum_mul_tsum_of_summable_norm if f and g are abolutely summable.
ℕ-indexed families (Cauchy product) #
We prove two versions of the Cauchy product formula. The first one is
tsum_mul_tsum_eq_tsum_sum_range, where the n-th term is a sum over finset.range (n+1)
involving nat substraction.
In order to avoid nat substraction, we also provide tsum_mul_tsum_eq_tsum_sum_antidiagonal,
where the n-th term is a sum over all pairs (k, l) such that k+l=n, which corresponds to the
finset finset.nat.antidiagonal n
theorem
summable_sum_mul_antidiagonal_of_summable_mul
{α : Type u_1}
[topological_space α]
[non_unital_non_assoc_semiring α]
[regular_space α]
[topological_semiring α]
{f g : ℕ → α}
(h : summable (λ (x : ℕ × ℕ), (f x.fst) * g x.snd)) :
theorem
tsum_mul_tsum_eq_tsum_sum_antidiagonal
{α : Type u_1}
{f g : ℕ → α}
[topological_space α]
[non_unital_non_assoc_semiring α]
[regular_space α]
[topological_semiring α]
(hf : summable f)
(hg : summable g)
(hfg : summable (λ (x : ℕ × ℕ), (f x.fst) * g x.snd)) :
The Cauchy product formula for the product of two infinites sums indexed by ℕ,
expressed by summing on finset.nat.antidiagonal.
See also tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
if f and g are absolutely summable.
theorem
summable_sum_mul_range_of_summable_mul
{α : Type u_1}
[topological_space α]
[non_unital_non_assoc_semiring α]
[regular_space α]
[topological_semiring α]
{f g : ℕ → α}
(h : summable (λ (x : ℕ × ℕ), (f x.fst) * g x.snd)) :
theorem
tsum_mul_tsum_eq_tsum_sum_range
{α : Type u_1}
{f g : ℕ → α}
[topological_space α]
[non_unital_non_assoc_semiring α]
[regular_space α]
[topological_semiring α]
(hf : summable f)
(hg : summable g)
(hfg : summable (λ (x : ℕ × ℕ), (f x.fst) * g x.snd)) :
The Cauchy product formula for the product of two infinites sums indexed by ℕ,
expressed by summing on finset.range.
See also tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm
if f and g are absolutely summable.