Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

[Merged by Bors] - feat(normed_space/inner_product): euclidean_space.norm_eq #6744

Closed
wants to merge 3 commits into from
Closed

[Merged by Bors] - feat(normed_space/inner_product): euclidean_space.norm_eq #6744

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
53756f0
feat(normed_space/inner_product): euclidean_space.norm_eq
pechersky Mar 16, 2021
f0c564c
add pi_Lp norm of nat and L2 norm lemmas
pechersky Mar 16, 2021
05ba1e0
Apply suggestions from code review
pechersky Mar 18, 2021
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Jump to
Jump to file
Failed to load files.
Diff view
Diff view
9 changes: 9 additions & 0 deletions src/analysis/normed_space/inner_product.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -1413,6 +1413,11 @@ instance pi_Lp.inner_product_space {ι : Type*} [fintype ι] (f : ι → Type*)
⟪x, y⟫ = ∑ i, ⟪x i, y i⟫ :=
rfl

lemma pi_Lp.norm_eq_of_L2 {ι : Type*} [fintype ι] {f : ι → Type*}
[Π i, inner_product_space 𝕜 (f i)] (x : pi_Lp 2 one_le_two f) :
∥x∥ = sqrt (∑ (i : ι), ∥x i∥ ^ 2) :=
by { rw [pi_Lp.norm_eq_of_nat 2]; simp [sqrt_eq_rpow] }

/-- A field `𝕜` satisfying `is_R_or_C` is itself a `𝕜`-inner product space. -/
instance is_R_or_C.inner_product_space : inner_product_space 𝕜 𝕜 :=
{ inner := (λ x y, (conj x) * y),
Expand All @@ -1430,6 +1435,10 @@ space use `euclidean_space 𝕜 (fin n)`. -/
def euclidean_space (𝕜 : Type*) [is_R_or_C 𝕜]
(n : Type*) [fintype n] : Type* := pi_Lp 2 one_le_two (λ (i : n), 𝕜)

lemma euclidean_space.norm_eq {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
(x : euclidean_space 𝕜 n) : ∥x∥ = real.sqrt (∑ (i : n), ∥x i∥ ^ 2) :=
pi_Lp.norm_eq_of_L2 x

/-! ### Inner product space structure on subspaces -/

/-- Induced inner product on a submodule. -/
Expand Down
5 changes: 5 additions & 0 deletions src/topology/metric_space/pi_Lp.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -229,6 +229,11 @@ lemma norm_eq {p : ℝ} {hp : 1 ≤ p} {α : ι → Type*}
[∀i, normed_group (α i)] (f : pi_Lp p hp α) :
∥f∥ = (∑ (i : ι), ∥f i∥ ^ p) ^ (1/p) := rfl

lemma norm_eq_of_nat {p : ℝ} {hp : 1 ≤ p} {α : ι → Type*}
Copy link
Collaborator Author

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

I added this lemma because of the notational confusion between ^ when encoding rpow or monoid.pow. For working with nat powers, this will allow for cleaner rewrites, with easy discharging with norm_num. I couldn't figure out how to use . tactic.norm_num as the out_param there.

[∀i, normed_group (α i)] (n : ℕ) (h : p = n) (f : pi_Lp p hp α) :
eric-wieser marked this conversation as resolved.
Show resolved Hide resolved
∥f∥ = (∑ (i : ι), ∥f i∥ ^ n) ^ (1/(n : ℝ)) :=
by simp [norm_eq, h, real.sqrt_eq_rpow, ←real.rpow_nat_cast]

variables (𝕜 : Type*) [normed_field 𝕜]

/-- The product of finitely many normed spaces is a normed space, with the `L^p` norm. -/
Expand Down