chore: split file on series of functions into two files (#9906)

Currently, the same file contains the facts that series of functions are continuous (resp. smooth) under suitable assumption. I will need the result on continuity in a file of more topological nature. To avoid importing all calculus in this file, this PR splits the file `Analysis.Calculus.Series` into `Analysis.Calculus.SmoothSeries` and `Analysis.NormedSpace.FunctionSeries`.

It's purely a splitting PR, no result added or removed.

Mathlib.lean

@@ -666,7 +666,7 @@ import Mathlib.Analysis.Calculus.Monotone
 import Mathlib.Analysis.Calculus.ParametricIntegral
 import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
 import Mathlib.Analysis.Calculus.Rademacher
-import Mathlib.Analysis.Calculus.Series
+import Mathlib.Analysis.Calculus.SmoothSeries
 import Mathlib.Analysis.Calculus.TangentCone
 import Mathlib.Analysis.Calculus.Taylor
 import Mathlib.Analysis.Calculus.UniformLimitsDeriv
@@ -833,6 +833,7 @@ import Mathlib.Analysis.NormedSpace.Exponential
 import Mathlib.Analysis.NormedSpace.Extend
 import Mathlib.Analysis.NormedSpace.Extr
 import Mathlib.Analysis.NormedSpace.FiniteDimension
+import Mathlib.Analysis.NormedSpace.FunctionSeries
 import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
 import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual
 import Mathlib.Analysis.NormedSpace.HahnBanach.Separation

Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean

@@ -3,7 +3,7 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Analysis.Calculus.Series
+import Mathlib.Analysis.Calculus.SmoothSeries
 import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
 import Mathlib.Analysis.Convolution
 import Mathlib.Analysis.InnerProductSpace.EuclideanDist

Mathlib/Analysis/Calculus/Series.lean → Mathlib/Analysis/Calculus/SmoothSeries.lean

@@ -8,17 +8,17 @@ import Mathlib.Analysis.Calculus.UniformLimitsDeriv
 import Mathlib.Data.Nat.Cast.WithTop
 import Mathlib.RingTheory.Ideal.LocalRing
 import Mathlib.Topology.Algebra.InfiniteSum.Module
+import Mathlib.Analysis.NormedSpace.FunctionSeries
 
 #align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
 
 /-!
 # Smoothness of series
 
-We show that series of functions are continuous, or differentiable, or smooth, when each individual
+We show that series of functions are differentiable, or smooth, when each individual
 function in the series is and additionally suitable uniform summable bounds are satisfied.
 
 More specifically,
-* `continuous_tsum` ensures that a series of continuous functions is continuous.
 * `differentiable_tsum` ensures that a series of differentiable functions is differentiable.
 * `contDiff_tsum` ensures that a series of smooth functions is smooth.
 
@@ -33,69 +33,6 @@ open scoped Topology NNReal BigOperators
 variable {α β 𝕜 E F : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
   [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ}
 
-/-! ### Continuity -/
-
-
-/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
-Version relative to a set, with general index set. -/
-theorem tendstoUniformlyOn_tsum {f : α → β → F} (hu : Summable u) {s : Set β}
-    (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
-    TendstoUniformlyOn (fun t : Finset α => fun x => ∑ n in t, f n x) (fun x => ∑' n, f n x) atTop
-      s := by
-  refine' tendstoUniformlyOn_iff.2 fun ε εpos => _
-  filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ εpos] with t ht x hx
-  have A : Summable fun n => ‖f n x‖ :=
-    .of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun n => hfu n x hx) hu
-  rw [dist_eq_norm, ← sum_add_tsum_subtype_compl A.of_norm t, add_sub_cancel']
-  apply lt_of_le_of_lt _ ht
-  apply (norm_tsum_le_tsum_norm (A.subtype _)).trans
-  exact tsum_le_tsum (fun n => hfu _ _ hx) (A.subtype _) (hu.subtype _)
-#align tendsto_uniformly_on_tsum tendstoUniformlyOn_tsum
-
-/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
-Version relative to a set, with index set `ℕ`. -/
-theorem tendstoUniformlyOn_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u) {s : Set β}
-    (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
-    TendstoUniformlyOn (fun N => fun x => ∑ n in Finset.range N, f n x) (fun x => ∑' n, f n x) atTop
-      s :=
-  fun v hv => tendsto_finset_range.eventually (tendstoUniformlyOn_tsum hu hfu v hv)
-#align tendsto_uniformly_on_tsum_nat tendstoUniformlyOn_tsum_nat
-
-/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
-Version with general index set. -/
-theorem tendstoUniformly_tsum {f : α → β → F} (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
-    TendstoUniformly (fun t : Finset α => fun x => ∑ n in t, f n x) (fun x => ∑' n, f n x) atTop :=
-  by rw [← tendstoUniformlyOn_univ]; exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x
-#align tendsto_uniformly_tsum tendstoUniformly_tsum
-
-/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
-Version with index set `ℕ`. -/
-theorem tendstoUniformly_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u)
-    (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
-    TendstoUniformly (fun N => fun x => ∑ n in Finset.range N, f n x) (fun x => ∑' n, f n x)
-      atTop :=
-  fun v hv => tendsto_finset_range.eventually (tendstoUniformly_tsum hu hfu v hv)
-#align tendsto_uniformly_tsum_nat tendstoUniformly_tsum_nat
-
-/-- An infinite sum of functions with summable sup norm is continuous on a set if each individual
-function is. -/
-theorem continuousOn_tsum [TopologicalSpace β] {f : α → β → F} {s : Set β}
-    (hf : ∀ i, ContinuousOn (f i) s) (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
-    ContinuousOn (fun x => ∑' n, f n x) s := by
-  classical
-    refine' (tendstoUniformlyOn_tsum hu hfu).continuousOn (eventually_of_forall _)
-    intro t
-    exact continuousOn_finset_sum _ fun i _ => hf i
-#align continuous_on_tsum continuousOn_tsum
-
-/-- An infinite sum of functions with summable sup norm is continuous if each individual
-function is. -/
-theorem continuous_tsum [TopologicalSpace β] {f : α → β → F} (hf : ∀ i, Continuous (f i))
-    (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) : Continuous fun x => ∑' n, f n x := by
-  simp_rw [continuous_iff_continuousOn_univ] at hf ⊢
-  exact continuousOn_tsum hf hu fun n x _ => hfu n x
-#align continuous_tsum continuous_tsum
-
 /-! ### Differentiability -/
 
 variable [NormedSpace 𝕜 F]

Mathlib/Analysis/Complex/LocallyUniformLimit.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 -/
 import Mathlib.Analysis.Complex.RemovableSingularity
-import Mathlib.Analysis.Calculus.Series
+import Mathlib.Analysis.Calculus.SmoothSeries
 
 #align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
 

Mathlib/Analysis/NormedSpace/FunctionSeries.lean

@@ -0,0 +1,84 @@
+/-
+Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sébastien Gouëzel
+-/
+import Mathlib.Analysis.Normed.Group.InfiniteSum
+import Mathlib.Topology.Instances.ENNReal
+
+#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
+/-!
+# Continuity of series of functions
+
+We show that series of functions are continuous when each individual function in the series is and
+additionally suitable uniform summable bounds are satisfied, in `continuous_tsum`.
+
+For smoothness of series of functions, see the file `Analysis.Calculus.SmoothSeries`.
+-/
+
+open Set Metric TopologicalSpace Function Filter
+
+open scoped Topology NNReal BigOperators
+
+variable {α β F : Type*} [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ}
+
+/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
+Version relative to a set, with general index set. -/
+theorem tendstoUniformlyOn_tsum {f : α → β → F} (hu : Summable u) {s : Set β}
+    (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
+    TendstoUniformlyOn (fun t : Finset α => fun x => ∑ n in t, f n x) (fun x => ∑' n, f n x) atTop
+      s := by
+  refine' tendstoUniformlyOn_iff.2 fun ε εpos => _
+  filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ εpos] with t ht x hx
+  have A : Summable fun n => ‖f n x‖ :=
+    .of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun n => hfu n x hx) hu
+  rw [dist_eq_norm, ← sum_add_tsum_subtype_compl A.of_norm t, add_sub_cancel']
+  apply lt_of_le_of_lt _ ht
+  apply (norm_tsum_le_tsum_norm (A.subtype _)).trans
+  exact tsum_le_tsum (fun n => hfu _ _ hx) (A.subtype _) (hu.subtype _)
+#align tendsto_uniformly_on_tsum tendstoUniformlyOn_tsum
+
+/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
+Version relative to a set, with index set `ℕ`. -/
+theorem tendstoUniformlyOn_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u) {s : Set β}
+    (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
+    TendstoUniformlyOn (fun N => fun x => ∑ n in Finset.range N, f n x) (fun x => ∑' n, f n x) atTop
+      s :=
+  fun v hv => tendsto_finset_range.eventually (tendstoUniformlyOn_tsum hu hfu v hv)
+#align tendsto_uniformly_on_tsum_nat tendstoUniformlyOn_tsum_nat
+
+/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
+Version with general index set. -/
+theorem tendstoUniformly_tsum {f : α → β → F} (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
+    TendstoUniformly (fun t : Finset α => fun x => ∑ n in t, f n x) (fun x => ∑' n, f n x) atTop :=
+  by rw [← tendstoUniformlyOn_univ]; exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x
+#align tendsto_uniformly_tsum tendstoUniformly_tsum
+
+/-- An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
+Version with index set `ℕ`. -/
+theorem tendstoUniformly_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u)
+    (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
+    TendstoUniformly (fun N => fun x => ∑ n in Finset.range N, f n x) (fun x => ∑' n, f n x)
+      atTop :=
+  fun v hv => tendsto_finset_range.eventually (tendstoUniformly_tsum hu hfu v hv)
+#align tendsto_uniformly_tsum_nat tendstoUniformly_tsum_nat
+
+/-- An infinite sum of functions with summable sup norm is continuous on a set if each individual
+function is. -/
+theorem continuousOn_tsum [TopologicalSpace β] {f : α → β → F} {s : Set β}
+    (hf : ∀ i, ContinuousOn (f i) s) (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
+    ContinuousOn (fun x => ∑' n, f n x) s := by
+  classical
+    refine' (tendstoUniformlyOn_tsum hu hfu).continuousOn (eventually_of_forall _)
+    intro t
+    exact continuousOn_finset_sum _ fun i _ => hf i
+#align continuous_on_tsum continuousOn_tsum
+
+/-- An infinite sum of functions with summable sup norm is continuous if each individual
+function is. -/
+theorem continuous_tsum [TopologicalSpace β] {f : α → β → F} (hf : ∀ i, Continuous (f i))
+    (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) : Continuous fun x => ∑' n, f n x := by
+  simp_rw [continuous_iff_continuousOn_univ] at hf ⊢
+  exact continuousOn_tsum hf hu fun n x _ => hfu n x
+#align continuous_tsum continuous_tsum