# `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)`.

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noncomputable instance WithLp.instProdInnerProductSpace {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] :
InnerProductSpace 𝕜 (WithLp 2 (E × F))
Equations

- One or more equations did not get rendered due to their size.

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@[simp]
theorem WithLp.prod_inner_apply {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] (x y : WithLp 2 (E × F)) :
inner 𝕜 x y = inner 𝕜 x.ofLp.1 y.ofLp.1 + inner 𝕜 x.ofLp.2 y.ofLp.2
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def OrthonormalBasis.prod {𝕜 : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [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

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@[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} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [Fintype ι₁] [Fintype ι₂] (v : OrthonormalBasis ι₁ 𝕜 E) (w : OrthonormalBasis ι₂ 𝕜 F) (i : ι₁ ⊕ ι₂) :
(v.prod w) i = Sum.elim (WithLp.toLp 2 ∘ ⇑(LinearMap.inl 𝕜 E F) ∘ ⇑v) (WithLp.toLp 2 ∘ ⇑(LinearMap.inr 𝕜 E F) ∘ ⇑w) i