Infinite sums in (semi)normed groups #

In a complete (semi)normed group,

summable_iff_vanishing_norm: a series ∑' i, f i is summable if and only if for any ε > 0, there exists a finite set s such that the sum ∑ i in t, f i over any finite set t disjoint with s has norm less than ε;
summable_of_norm_bounded, summable_of_norm_bounded_eventually: if ∥f i∥ is bounded above by a summable series ∑' i, g i, then ∑' i, f i is summable as well; the same is true if the inequality hold only off some finite set.
tsum_of_norm_bounded, has_sum.norm_le_of_bounded: if ∥f i∥ ≤ g i, where ∑' i, g i is a summable series, then ∥∑' i, f i∥ ≤ ∑' i, g i.

Tags #

infinite series, absolute convergence, normed group

theorem cauchy_seq_finset_iff_vanishing_norm {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} :

cauchy_seq (λ (s : finset ι), ∑ (i : ι) in s, f i) ↔ ∀ (ε : ℝ), ε > 0 → (∃ (s : finset ι), ∀ (t : finset ι), disjoint t s → ∥∑ (i : ι) in t, f i∥ < ε)

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theorem summable_iff_vanishing_norm {ι : Type u_1} {E : Type u_3} [semi_normed_group E] [complete_space E] {f : ι → E} :

summable f ↔ ∀ (ε : ℝ), ε > 0 → (∃ (s : finset ι), ∀ (t : finset ι), disjoint t s → ∥∑ (i : ι) in t, f i∥ < ε)

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theorem cauchy_seq_finset_of_norm_bounded_eventually {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} {g : ι → ℝ} (hg : summable g) (h : ∀ᶠ (i : ι) in filter.cofinite, ∥f i∥ ≤ g i) :

cauchy_seq (λ (s : finset ι), ∑ (i : ι) in s, f i)

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theorem cauchy_seq_finset_of_norm_bounded {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} (g : ι → ℝ) (hg : summable g) (h : ∀ (i : ι), ∥f i∥ ≤ g i) :

cauchy_seq (λ (s : finset ι), ∑ (i : ι) in s, f i)

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theorem cauchy_seq_finset_of_summable_norm {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} (hf : summable (λ (a : ι), ∥f a∥)) :

cauchy_seq (λ (s : finset ι), ∑ (a : ι) in s, f a)

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theorem has_sum_of_subseq_of_summable {ι : Type u_1} {α : Type u_2} {E : Type u_3} [semi_normed_group E] {f : ι → E} (hf : summable (λ (a : ι), ∥f a∥)) {s : α → finset ι} {p : filter α} [p.ne_bot] (hs : filter.tendsto s p filter.at_top) {a : E} (ha : filter.tendsto (λ (b : α), ∑ (i : ι) in s b, f i) p (𝓝 a)) :

has_sum f a

If a function f is summable in norm, and along some sequence of finsets exhausting the space its sum is converging to a limit a, then this holds along all finsets, i.e., f is summable with sum a.

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theorem has_sum_iff_tendsto_nat_of_summable_norm {E : Type u_3} [semi_normed_group E] {f : ℕ → E} {a : E} (hf : summable (λ (i : ℕ), ∥f i∥)) :

has_sum f a ↔ filter.tendsto (λ (n : ℕ), ∑ (i : ℕ) in finset.range n, f i) filter.at_top (𝓝 a)

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theorem summable_of_norm_bounded {ι : Type u_1} {E : Type u_3} [semi_normed_group E] [complete_space E] {f : ι → E} (g : ι → ℝ) (hg : summable g) (h : ∀ (i : ι), ∥f i∥ ≤ g i) :

summable f

The direct comparison test for series: if the norm of f is bounded by a real function g which is summable, then f is summable.

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theorem has_sum.norm_le_of_bounded {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} {g : ι → ℝ} {a : E} {b : ℝ} (hf : has_sum f a) (hg : has_sum g b) (h : ∀ (i : ι), ∥f i∥ ≤ g i) :

∥a∥ ≤ b

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theorem tsum_of_norm_bounded {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : has_sum g a) (h : ∀ (i : ι), ∥f i∥ ≤ g i) :

∥∑' (i : ι), f i∥ ≤ a

Quantitative result associated to the direct comparison test for series: If ∑' i, g i is summable, and for all i, ∥f i∥ ≤ g i, then ∥∑' i, f i∥ ≤ ∑' i, g i. Note that we do not assume that ∑' i, f i is summable, and it might not be the case if α is not a complete space.

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theorem norm_tsum_le_tsum_norm {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} (hf : summable (λ (i : ι), ∥f i∥)) :

∥∑' (i : ι), f i∥ ≤ ∑' (i : ι), ∥f i∥

If ∑' i, ∥f i∥ is summable, then ∥∑' i, f i∥ ≤ (∑' i, ∥f i∥). Note that we do not assume that ∑' i, f i is summable, and it might not be the case if α is not a complete space.

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theorem tsum_of_nnnorm_bounded {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} {g : ι → ℝ≥0} {a : ℝ≥0} (hg : has_sum g a) (h : ∀ (i : ι), ∥f i∥₊ ≤ g i) :

∥∑' (i : ι), f i∥₊ ≤ a

Quantitative result associated to the direct comparison test for series: If ∑' i, g i is summable, and for all i, ∥f i∥₊ ≤ g i, then ∥∑' i, f i∥₊ ≤ ∑' i, g i. Note that we do not assume that ∑' i, f i is summable, and it might not be the case if α is not a complete space.

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theorem nnnorm_tsum_le {ι : Type u_1} {E : Type u_3} [semi_normed_group E] {f : ι → E} (hf : summable (λ (i : ι), ∥f i∥₊)) :

∥∑' (i : ι), f i∥₊ ≤ ∑' (i : ι), ∥f i∥₊

If ∑' i, ∥f i∥₊ is summable, then ∥∑' i, f i∥₊ ≤ ∑' i, ∥f i∥₊. Note that we do not assume that ∑' i, f i is summable, and it might not be the case if α is not a complete space.

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theorem summable_of_norm_bounded_eventually {ι : Type u_1} {E : Type u_3} [semi_normed_group E] [complete_space E] {f : ι → E} (g : ι → ℝ) (hg : summable g) (h : ∀ᶠ (i : ι) in filter.cofinite, ∥f i∥ ≤ g i) :

summable f

Variant of the direct comparison test for series: if the norm of f is eventually bounded by a real function g which is summable, then f is summable.

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theorem summable_of_nnnorm_bounded {ι : Type u_1} {E : Type u_3} [semi_normed_group E] [complete_space E] {f : ι → E} (g : ι → ℝ≥0) (hg : summable g) (h : ∀ (i : ι), ∥f i∥₊ ≤ g i) :

summable f

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theorem summable_of_summable_norm {ι : Type u_1} {E : Type u_3} [semi_normed_group E] [complete_space E] {f : ι → E} (hf : summable (λ (a : ι), ∥f a∥)) :

summable f

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theorem summable_of_summable_nnnorm {ι : Type u_1} {E : Type u_3} [semi_normed_group E] [complete_space E] {f : ι → E} (hf : summable (λ (a : ι), ∥f a∥₊)) :

summable f