# L² inner product space structure on products of inner product spaces #

The L² norm on product of two inner product spaces is compatible with an inner product $$\langle x, y\rangle = \langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle.$$ This is recorded in this file as an inner product space instance on WithLp 2 (E × F).

noncomputable instance WithLp.instProdInnerProductSpace {𝕜 : Type u_1} (E : Type u_4) (F : Type u_5) [] [] [] :
InnerProductSpace 𝕜 (WithLp 2 (E × F))
Equations
@[simp]
theorem WithLp.prod_inner_apply {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [] [] [] (x : WithLp 2 (E × F)) (y : WithLp 2 (E × F)) :
x, y⟫_𝕜 = x.1, y.1⟫_𝕜 + x.2, y.2⟫_𝕜
def OrthonormalBasis.prod {𝕜 : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} {E : Type u_4} {F : Type u_5} [] [] [] [Fintype ι₁] [Fintype ι₂] (v : OrthonormalBasis ι₁ 𝕜 E) (w : OrthonormalBasis ι₂ 𝕜 F) :
OrthonormalBasis (ι₁ ι₂) 𝕜 (WithLp 2 (E × F))

The product of two orthonormal bases is a basis for the L2-product.

Equations
• v.prod w = ((v.toBasis.prod w.toBasis).map (WithLp.linearEquiv 2 𝕜 (E × F)).symm).toOrthonormalBasis
Instances For
@[simp]
theorem OrthonormalBasis.prod_apply {𝕜 : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} {E : Type u_4} {F : Type u_5} [] [] [] [Fintype ι₁] [Fintype ι₂] (v : OrthonormalBasis ι₁ 𝕜 E) (w : OrthonormalBasis ι₂ 𝕜 F) (i : ι₁ ι₂) :
(v.prod w) i = Sum.elim ((LinearMap.inl 𝕜 E F) v) ((LinearMap.inr 𝕜 E F) w) i