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

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

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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
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9 changes: 9 additions & 0 deletions src/analysis/normed_space/inner_product.lean
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Expand Up @@ -1430,6 +1430,15 @@ 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) :=
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begin
have : (2 : ℝ) = (2 : ℕ) := by norm_num,
have h : ∀ (x : ℝ) (n : ℕ), real.rpow x n = monoid.pow x n := by simp,
simp_rw [pi_Lp.norm_eq, real.sqrt_eq_rpow, this, ←real.rpow_eq_pow, h],
simp
end
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/-! ### Inner product space structure on subspaces -/

/-- Induced inner product on a submodule. -/
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