mathlib documentation

analysis.normed.group.infinite_sum

Infinite sums in (semi)normed groups #

In a complete (semi)normed group,

Tags #

infinite series, absolute convergence, normed group

theorem cauchy_seq_finset_iff_vanishing_norm {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} :
cauchy_seq (λ (s : finset ι), s.sum (λ (i : ι), f i)) (ε : ), ε > 0 ( (s : finset ι), (t : finset ι), disjoint t s t.sum (λ (i : ι), f i) < ε)
theorem summable_iff_vanishing_norm {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] [complete_space E] {f : ι E} :
summable f (ε : ), ε > 0 ( (s : finset ι), (t : finset ι), disjoint t s t.sum (λ (i : ι), f i) < ε)
theorem cauchy_seq_finset_of_norm_bounded_eventually {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} {g : ι } (hg : summable g) (h : ∀ᶠ (i : ι) in filter.cofinite, f i g i) :
cauchy_seq (λ (s : finset ι), s.sum (λ (i : ι), f i))
theorem cauchy_seq_finset_of_norm_bounded {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} (g : ι ) (hg : summable g) (h : (i : ι), f i g i) :
cauchy_seq (λ (s : finset ι), s.sum (λ (i : ι), f i))
theorem cauchy_seq_range_of_norm_bounded {E : Type u_3} [seminormed_add_comm_group E] {f : E} (g : ) (hg : cauchy_seq (λ (n : ), (finset.range n).sum (λ (i : ), g i))) (hf : (i : ), f i g i) :
cauchy_seq (λ (n : ), (finset.range n).sum (λ (i : ), f i))

A version of the direct comparison test for conditionally convergent series. See cauchy_seq_finset_of_norm_bounded for the same statement about absolutely convergent ones.

theorem cauchy_seq_finset_of_summable_norm {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} (hf : summable (λ (a : ι), f a)) :
cauchy_seq (λ (s : finset ι), s.sum (λ (a : ι), f a))
theorem has_sum_of_subseq_of_summable {ι : Type u_1} {α : Type u_2} {E : Type u_3} [seminormed_add_comm_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), (s b).sum (λ (i : ι), f i)) p (nhds 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.

theorem has_sum_iff_tendsto_nat_of_summable_norm {E : Type u_3} [seminormed_add_comm_group E] {f : E} {a : E} (hf : summable (λ (i : ), f i)) :
has_sum f a filter.tendsto (λ (n : ), (finset.range n).sum (λ (i : ), f i)) filter.at_top (nhds a)
theorem summable_of_norm_bounded {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] [complete_space E] {f : ι E} (g : ι ) (hg : summable g) (h : (i : ι), f i g i) :

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.

theorem has_sum.norm_le_of_bounded {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} {g : ι } {a : E} {b : } (hf : has_sum f a) (hg : has_sum g b) (h : (i : ι), f i g i) :
theorem tsum_of_norm_bounded {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} {g : ι } {a : } (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.

theorem norm_tsum_le_tsum_norm {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_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.

theorem tsum_of_nnnorm_bounded {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] {f : ι E} {g : ι nnreal} {a : nnreal} (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.

theorem nnnorm_tsum_le {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_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.

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

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.

theorem summable_of_nnnorm_bounded {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] [complete_space E] {f : ι E} (g : ι nnreal) (hg : summable g) (h : (i : ι), f i‖₊ g i) :
theorem summable_of_summable_norm {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] [complete_space E] {f : ι E} (hf : summable (λ (a : ι), f a)) :
theorem summable_of_summable_nnnorm {ι : Type u_1} {E : Type u_3} [seminormed_add_comm_group E] [complete_space E] {f : ι E} (hf : summable (λ (a : ι), f a‖₊)) :