wip tensor inner product

src/for_mathlib/linear_algebra/bilinear_form/tensor_product.lean

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+/-
+Copyright (c) 2023 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+-/
+import linear_algebra.bilinear_form.tensor_product
+
+universes u v w
+variables {ι : Type*} {R : Type*} {M₁ M₂ : Type*}
+
+open_locale tensor_product
+
+namespace bilin_form
+
+section comm_semiring
+variables [comm_semiring R]
+variables [add_comm_monoid M₁] [add_comm_monoid M₂]
+variables [module R M₁] [module R M₂]
+
+lemma is_symm.tmul {B₁ : bilin_form R M₁} {B₂ : bilin_form R M₂}
+  (hB₁ : B₁.is_symm) (hB₂ : B₂.is_symm) : (B₁.tmul B₂).is_symm :=
+suffices (B₁.tmul B₂).to_lin = (B₁.tmul B₂).to_lin.flip,
+begin
+  intros x y,
+  have := fun_like.congr_fun (fun_like.congr_fun this x) y,
+  exact this
+end,
+tensor_product.ext' $ λ x₁ x₂, tensor_product.ext' $ λ y₁ y₂,
+  (congr_arg2 (*) (hB₁ x₁ y₁) (hB₂ x₂ y₂) : _)
+
+end comm_semiring
+
+end bilin_form

src/for_mathlib/linear_algebra/tensor_product/inner_product_space.lean

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+import for_mathlib.linear_algebra.bilinear_form.tensor_product
+
+import analysis.inner_product_space.basic
+
+variables {V W : Type*} [normed_add_comm_group V] [normed_add_comm_group W]
+variables [inner_product_space ℝ V] [inner_product_space ℝ W]
+
+open_locale tensor_product
+
+noncomputable instance : inner_product_space.core ℝ (V ⊗[ℝ] W) :=
+{ inner := λ x y, bilin_form_of_real_inner.tmul bilin_form_of_real_inner x y,
+  conj_symm := λ x y,
+    bilin_form.is_symm.tmul (λ x y, real_inner_comm _ _) (λ x y, real_inner_comm _ _) y x,
+  nonneg_re := λ x, begin
+    simp only [is_R_or_C.re_to_real],
+    induction x using tensor_product.induction_on with v w x y hx hy,
+    { simp only [bilin_form.zero_right] },
+    { simp only [bilin_form.tmul.equations._eqn_1, bilin_form_of_real_inner_apply,
+        bilin_form.tensor_distrib_tmul],
+      exact mul_nonneg real_inner_self_nonneg real_inner_self_nonneg, },
+    { simp only [bilin_form.add_left, bilin_form.add_right],
+      sorry },
+  end,
+  definite := sorry,
+  add_left := bilin_form.add_left,
+  smul_left := λ _ _ _, bilin_form.bilin_smul_left _ _ _ _ }
+
+noncomputable instance : normed_add_comm_group (V ⊗[ℝ] W) :=
+@inner_product_space.core.to_normed_add_comm_group ℝ _ _ _ _ _
+
+noncomputable instance : inner_product_space ℝ (V ⊗[ℝ] W) :=
+inner_product_space.of_core _
+
+@[simp] lemma tmul_inner_tmul (v₁ : V) (w₁ : W) (v₂ : V) (w₂ : W) :
+  @inner ℝ _ _ (v₁ ⊗ₜ[ℝ] w₁) (v₂ ⊗ₜ[ℝ] w₂) = inner v₁ v₂ * inner w₁ w₂ :=
+bilin_form.tensor_distrib_tmul _ _ _ _ _ _
+
+@[simp] lemma norm_tmul (v : V) (w : W) : ‖v ⊗ₜ[ℝ] w‖ = ‖v‖ * ‖w‖ :=
+begin
+  have := congr_arg real.sqrt (tmul_inner_tmul v w v w),
+  simp_rw real_inner_self_eq_norm_sq at this,
+  rw [← mul_pow, real.sqrt_sq (norm_nonneg _),
+    real.sqrt_sq (mul_nonneg (norm_nonneg _) (norm_nonneg _))] at this,
+  exact this,
+end
+
+@[simp] lemma nnnorm_tmul (v : V) (w : W) : ‖v ⊗ₜ[ℝ] w‖₊ = ‖v‖₊ * ‖w‖₊ :=
+nnreal.eq $ norm_tmul _ _