File: Mathlib/Analysis/Normed/Group/InfiniteSum.lean

full_name: cauchySeq_finset_iff_vanishing_norm
code: theorem cauchySeq_finset_iff_vanishing_norm {f : ι → E} :
    (CauchySeq fun s : Finset ι => ∑ i ∈ s, f i) ↔
      ∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε 
goal: (CauchySeq fun s => ∑ i ∈ s, f i) ↔ ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε

full_name: summable_iff_vanishing_norm
code: theorem summable_iff_vanishing_norm [CompleteSpace E] {f : ι → E} :
    Summable f ↔ ∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε 
goal: Summable f ↔ ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε

full_name: cauchySeq_finset_of_norm_bounded_eventually
code: theorem cauchySeq_finset_of_norm_bounded_eventually {f : ι → E} {g : ι → ℝ} (hg : Summable g)
    (h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : CauchySeq fun s => ∑ i ∈ s, f i 
goal: CauchySeq fun s => ∑ i ∈ s, f i

full_name: cauchySeq_finset_of_norm_bounded
code: theorem cauchySeq_finset_of_norm_bounded {f : ι → E} (g : ι → ℝ) (hg : Summable g)
    (h : ∀ i, ‖f i‖ ≤ g i) : CauchySeq fun s : Finset ι => ∑ i ∈ s, f i 
goal: CauchySeq fun s => ∑ i ∈ s, f i

full_name: cauchySeq_range_of_norm_bounded
code: /-- A version of the **direct comparison test** for conditionally convergent series.
See `cauchySeq_finset_of_norm_bounded` for the same statement about absolutely convergent ones. -/
theorem cauchySeq_range_of_norm_bounded {f : ℕ → E} (g : ℕ → ℝ)
    (hg : CauchySeq fun n => ∑ i ∈ range n, g i) (hf : ∀ i, ‖f i‖ ≤ g i) :
    CauchySeq fun n => ∑ i ∈ range n, f i 
goal: CauchySeq fun n => ∑ i ∈ Finset.range n, f i

full_name: cauchySeq_finset_of_summable_norm
code: theorem cauchySeq_finset_of_summable_norm {f : ι → E} (hf : Summable fun a => ‖f a‖) :
    CauchySeq fun s : Finset ι => ∑ a ∈ s, f a 
goal: CauchySeq fun s => ∑ a ∈ s, f a

full_name: hasSum_of_subseq_of_summable
code: /-- 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 hasSum_of_subseq_of_summable {f : ι → E} (hf : Summable fun a => ‖f a‖) {s : α → Finset ι}
    {p : Filter α} [NeBot p] (hs : Tendsto s p atTop) {a : E}
    (ha : Tendsto (fun b => ∑ i ∈ s b, f i) p (𝓝 a)) : HasSum f a 
goal: HasSum f a

full_name: hasSum_iff_tendsto_nat_of_summable_norm
code: theorem hasSum_iff_tendsto_nat_of_summable_norm {f : ℕ → E} {a : E} (hf : Summable fun i => ‖f i‖) :
    HasSum f a ↔ Tendsto (fun n : ℕ => ∑ i ∈ range n, f i) atTop (𝓝 a) 
goal: HasSum f a ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 a)

full_name: Summable.of_norm_bounded
code: /-- 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 Summable.of_norm_bounded [CompleteSpace E] {f : ι → E} (g : ι → ℝ) (hg : Summable g)
    (h : ∀ i, ‖f i‖ ≤ g i) : Summable f 
goal: Summable f

full_name: HasSum.norm_le_of_bounded
code: theorem HasSum.norm_le_of_bounded {f : ι → E} {g : ι → ℝ} {a : E} {b : ℝ} (hf : HasSum f a)
    (hg : HasSum g b) (h : ∀ i, ‖f i‖ ≤ g i) : ‖a‖ ≤ b 
goal: ‖a‖ ≤ b

full_name: tsum_of_norm_bounded
code: /-- 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 tsum_of_norm_bounded {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : HasSum g a)
    (h : ∀ i, ‖f i‖ ≤ g i) : ‖∑' i : ι, f i‖ ≤ a 
goal: ‖∑' (i : ι), f i‖ ≤ a

full_name: norm_tsum_le_tsum_norm
code: /-- 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 norm_tsum_le_tsum_norm {f : ι → E} (hf : Summable fun i => ‖f i‖) :
    ‖∑' i, f i‖ ≤ ∑' i, ‖f i‖ 
goal: ‖∑' (i : ι), f i‖ ≤ ∑' (i : ι), ‖f i‖

full_name: tsum_of_nnnorm_bounded
code: /-- 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 tsum_of_nnnorm_bounded {f : ι → E} {g : ι → ℝ≥0} {a : ℝ≥0} (hg : HasSum g a)
    (h : ∀ i, ‖f i‖₊ ≤ g i) : ‖∑' i : ι, f i‖₊ ≤ a 
goal: ‖∑' (i : ι), f i‖₊ ≤ a

full_name: nnnorm_tsum_le
code: /-- 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 nnnorm_tsum_le {f : ι → E} (hf : Summable fun i => ‖f i‖₊) : ‖∑' i, f i‖₊ ≤ ∑' i, ‖f i‖₊ 
goal: ‖∑' (i : ι), f i‖₊ ≤ ∑' (i : ι), ‖f i‖₊

full_name: Summable.of_norm_bounded_eventually
code: /-- 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_norm_bounded_eventually {f : ι → E} (g : ι → ℝ) (hg : Summable g)
    (h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : Summable f 
goal: Summable f

full_name: Summable.of_norm_bounded_eventually_nat
code: /-- 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_norm_bounded_eventually_nat {f : ℕ → E} (g : ℕ → ℝ) (hg : Summable g)
    (h : ∀ᶠ i in atTop, ‖f i‖ ≤ g i) : Summable f 
goal: Summable f

full_name: Summable.of_nnnorm_bounded
code: theorem Summable.of_nnnorm_bounded {f : ι → E} (g : ι → ℝ≥0) (hg : Summable g)
    (h : ∀ i, ‖f i‖₊ ≤ g i) : Summable f 
goal: Summable f

full_name: Summable.of_norm
code: theorem Summable.of_norm {f : ι → E} (hf : Summable fun a => ‖f a‖) : Summable f 
goal: Summable f

full_name: Summable.of_nnnorm
code: theorem Summable.of_nnnorm {f : ι → E} (hf : Summable fun a => ‖f a‖₊) : Summable f 
goal: Summable f