statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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tendsto_integral_exp_inner_smul_cocompact :
tendsto (λ w : V, ∫ v, e [-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) | begin
by_cases hfi : integrable f, swap,
{ convert tendsto_const_nhds,
ext1 w,
apply integral_undef,
rwa ←fourier_integrand_integrable w },
refine metric.tendsto_nhds.mpr (λ ε hε, _),
obtain ⟨g, hg_supp, hfg, hg_cont, -⟩ :=
hfi.exists_has_compact_support_integral_sub_le (div_pos hε two_pos),
r... | theorem | tendsto_integral_exp_inner_smul_cocompact | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"add_halves",
"div_pos",
"fourier_integrand_integrable",
"tendsto_const_nhds",
"tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support",
"vector_fourier.norm_fourier_integral_le_integral_norm"
] | Riemann-Lebesgue lemma for functions on a real inner-product space: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 as `w → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.tendsto_integral_exp_smul_cocompact (f : ℝ → E) :
tendsto (λ w : ℝ, ∫ v : ℝ, e [-(v * w)] • f v) (cocompact ℝ) (𝓝 0) | tendsto_integral_exp_inner_smul_cocompact f | lemma | real.tendsto_integral_exp_smul_cocompact | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"tendsto_integral_exp_inner_smul_cocompact"
] | The Riemann-Lebesgue lemma for functions on `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.zero_at_infty_fourier_integral (f : ℝ → E) :
tendsto (𝓕 f) (cocompact ℝ) (𝓝 0) | tendsto_integral_exp_inner_smul_cocompact f | theorem | real.zero_at_infty_fourier_integral | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"tendsto_integral_exp_inner_smul_cocompact"
] | The Riemann-Lebesgue lemma for functions on `ℝ`, formulated via `real.fourier_integral`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_exp_smul_cocompact_of_inner_product (μ : measure V) [μ.is_add_haar_measure] :
tendsto (λ w : V →L[ℝ] ℝ, ∫ v, e[-w v] • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) | begin
obtain ⟨C, C_ne_zero, C_ne_top, hC⟩ := μ.is_add_haar_measure_eq_smul_is_add_haar_measure volume,
rw hC,
simp_rw integral_smul_measure,
rw ←(smul_zero _ : C.to_real • (0 : E) = 0),
apply tendsto.const_smul,
let A := (inner_product_space.to_dual ℝ V).symm,
have : (λ w : V →L[ℝ] ℝ, ∫ v, e[-w v] • f v) ... | lemma | tendsto_integral_exp_smul_cocompact_of_inner_product | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"inner_product_space.to_dual",
"inner_product_space.to_dual_symm_apply",
"is_R_or_C.conj_to_real",
"real.fourier_char_apply",
"smul_zero",
"tendsto_integral_exp_inner_smul_cocompact"
] | Riemann-Lebesgue lemma for functions on a finite-dimensional inner-product space, formulated
via dual space. **Do not use** -- it is only a stepping stone to
`tendsto_integral_exp_smul_cocompact` where the inner-product-space structure isn't required. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_exp_smul_cocompact (μ : measure V) [μ.is_add_haar_measure] :
tendsto (λ w : V →L[ℝ] ℝ, ∫ v, e[-w v] • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) | begin
-- We have already proved the result for inner-product spaces, formulated in a way which doesn't
-- refer to the inner product. So we choose an arbitrary inner-product space isomorphic to V
-- and port the result over from there.
let V' := euclidean_space ℝ (fin (finrank ℝ V)),
have A : V ≃L[ℝ] V' := to... | theorem | tendsto_integral_exp_smul_cocompact | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"continuous_linear_equiv.apply_symm_apply",
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.coe_def_rev",
"continuous_linear_equiv.symm_apply_apply",
"continuous_linear_map.add_apply",
"continuous_linear_map.coe_comp'",
"continuous_linear_map.smul_apply",
"euclidean_space",
"inv_fun",
... | Riemann-Lebesgue lemma for functions on a finite-dimensional real vector space, formulated via
dual space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.zero_at_infty_vector_fourier_integral (μ : measure V) [μ.is_add_haar_measure] :
tendsto (vector_fourier.fourier_integral e μ (top_dual_pairing ℝ V).flip f)
(cocompact (V →L[ℝ] ℝ)) (𝓝 0) | tendsto_integral_exp_smul_cocompact f μ | theorem | real.zero_at_infty_vector_fourier_integral | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"tendsto_integral_exp_smul_cocompact",
"top_dual_pairing",
"vector_fourier.fourier_integral"
] | The Riemann-Lebesgue lemma, formulated in terms of `vector_fourier.fourier_integral` (with the
pairing in the definition of `fourier_integral` taken to be the canonical pairing between `V` and
its dual space). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_aux : (E →L[𝕜] F) →L⋆[𝕜] (F →L[𝕜] E) | (continuous_linear_map.compSL _ _ _ _ _ ((to_dual 𝕜 E).symm : normed_space.dual 𝕜 E →L⋆[𝕜] E)).comp
(to_sesq_form : (E →L[𝕜] F) →L[𝕜] F →L⋆[𝕜] normed_space.dual 𝕜 E) | def | continuous_linear_map.adjoint_aux | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.compSL",
"normed_space.dual"
] | The adjoint, as a continuous conjugate-linear map. This is only meant as an auxiliary
definition for the main definition `adjoint`, where this is bundled as a conjugate-linear isometric
equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_aux_apply (A : E →L[𝕜] F) (x : F) :
adjoint_aux A x = ((to_dual 𝕜 E).symm : (normed_space.dual 𝕜 E) → E) ((to_sesq_form A) x) | rfl | lemma | continuous_linear_map.adjoint_aux_apply | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"normed_space.dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_aux_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪adjoint_aux A y, x⟫ = ⟪y, A x⟫ | by { simp only [adjoint_aux_apply, to_dual_symm_apply, to_sesq_form_apply_coe, coe_comp',
innerSL_apply_coe]} | lemma | continuous_linear_map.adjoint_aux_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"innerSL_apply_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_aux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) : ⟪x, adjoint_aux A y⟫ = ⟪A x, y⟫ | by rw [←inner_conj_symm, adjoint_aux_inner_left, inner_conj_symm] | lemma | continuous_linear_map.adjoint_aux_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"inner_conj_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_aux_adjoint_aux (A : E →L[𝕜] F) : adjoint_aux (adjoint_aux A) = A | begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
rw [adjoint_aux_inner_right, adjoint_aux_inner_left],
end | lemma | continuous_linear_map.adjoint_aux_adjoint_aux | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"ext_inner_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_aux_norm (A : E →L[𝕜] F) : ‖adjoint_aux A‖ = ‖A‖ | begin
refine le_antisymm _ _,
{ refine continuous_linear_map.op_norm_le_bound _ (norm_nonneg _) (λ x, _),
rw [adjoint_aux_apply, linear_isometry_equiv.norm_map],
exact to_sesq_form_apply_norm_le },
{ nth_rewrite_lhs 0 [←adjoint_aux_adjoint_aux A],
refine continuous_linear_map.op_norm_le_bound _ (norm_... | lemma | continuous_linear_map.adjoint_aux_norm | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.op_norm_le_bound",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint : (E →L[𝕜] F) ≃ₗᵢ⋆[𝕜] (F →L[𝕜] E) | linear_isometry_equiv.of_surjective
{ norm_map' := adjoint_aux_norm,
..adjoint_aux }
(λ A, ⟨adjoint_aux A, adjoint_aux_adjoint_aux A⟩) | def | continuous_linear_map.adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"linear_isometry_equiv.of_surjective"
] | The adjoint of a bounded operator from Hilbert space E to Hilbert space F. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪A† y, x⟫ = ⟪y, A x⟫ | adjoint_aux_inner_left A x y | lemma | continuous_linear_map.adjoint_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | The fundamental property of the adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_inner_right (A : E →L[𝕜] F) (x : E) (y : F) : ⟪x, A† y⟫ = ⟪A x, y⟫ | adjoint_aux_inner_right A x y | lemma | continuous_linear_map.adjoint_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | The fundamental property of the adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_adjoint (A : E →L[𝕜] F) : A†† = A | adjoint_aux_adjoint_aux A | lemma | continuous_linear_map.adjoint_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | The adjoint is involutive | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_comp (A : F →L[𝕜] G) (B : E →L[𝕜] F) : (A ∘L B)† = B† ∘L A† | begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
simp only [adjoint_inner_right, continuous_linear_map.coe_comp', function.comp_app],
end | lemma | continuous_linear_map.adjoint_comp | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.coe_comp'",
"ext_inner_left"
] | The adjoint of the composition of two operators is the composition of the two adjoints
in reverse order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_norm_sq_eq_inner_adjoint_left (A : E →L[𝕜] E) (x : E) : ‖A x‖^2 = re ⟪(A† * A) x, x⟫ | have h : ⟪(A† * A) x, x⟫ = ⟪A x, A x⟫ := by { rw [←adjoint_inner_left], refl },
by rw [h, ←inner_self_eq_norm_sq _] | lemma | continuous_linear_map.apply_norm_sq_eq_inner_adjoint_left | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_norm_eq_sqrt_inner_adjoint_left (A : E →L[𝕜] E) (x : E) :
‖A x‖ = real.sqrt (re ⟪(A† * A) x, x⟫) | by rw [←apply_norm_sq_eq_inner_adjoint_left, real.sqrt_sq (norm_nonneg _)] | lemma | continuous_linear_map.apply_norm_eq_sqrt_inner_adjoint_left | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"real.sqrt",
"real.sqrt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_norm_sq_eq_inner_adjoint_right (A : E →L[𝕜] E) (x : E) : ‖A x‖^2 = re ⟪x, (A† * A) x⟫ | have h : ⟪x, (A† * A) x⟫ = ⟪A x, A x⟫ := by { rw [←adjoint_inner_right], refl },
by rw [h, ←inner_self_eq_norm_sq _] | lemma | continuous_linear_map.apply_norm_sq_eq_inner_adjoint_right | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_norm_eq_sqrt_inner_adjoint_right (A : E →L[𝕜] E) (x : E) :
‖A x‖ = real.sqrt (re ⟪x, (A† * A) x⟫) | by rw [←apply_norm_sq_eq_inner_adjoint_right, real.sqrt_sq (norm_nonneg _)] | lemma | continuous_linear_map.apply_norm_eq_sqrt_inner_adjoint_right | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"real.sqrt",
"real.sqrt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_adjoint_iff (A : E →L[𝕜] F) (B : F →L[𝕜] E) :
A = B† ↔ (∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫) | begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
ext x,
exact ext_inner_right 𝕜 (λ y, by simp only [adjoint_inner_left, h x y])
end | lemma | continuous_linear_map.eq_adjoint_iff | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"ext_inner_right"
] | The adjoint is unique: a map `A` is the adjoint of `B` iff it satisfies `⟪A x, y⟫ = ⟪x, B y⟫`
for all `x` and `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_id : (continuous_linear_map.id 𝕜 E).adjoint = continuous_linear_map.id 𝕜 E | begin
refine eq.symm _,
rw eq_adjoint_iff,
simp,
end | lemma | continuous_linear_map.adjoint_id | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.adjoint_subtypeL (U : submodule 𝕜 E)
[complete_space U] :
(U.subtypeL)† = orthogonal_projection U | begin
symmetry,
rw eq_adjoint_iff,
intros x u,
rw [U.coe_inner, inner_orthogonal_projection_left_eq_right,
orthogonal_projection_mem_subspace_eq_self],
refl
end | lemma | submodule.adjoint_subtypeL | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"inner_orthogonal_projection_left_eq_right",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_eq_self",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.adjoint_orthogonal_projection (U : submodule 𝕜 E)
[complete_space U] :
(orthogonal_projection U : E →L[𝕜] U)† = U.subtypeL | by rw [← U.adjoint_subtypeL, adjoint_adjoint] | lemma | submodule.adjoint_orthogonal_projection | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"orthogonal_projection",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_adjoint (A : E →L[𝕜] E) : star A = A† | rfl | lemma | continuous_linear_map.star_eq_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint_iff' {A : E →L[𝕜] E} : is_self_adjoint A ↔ A.adjoint = A | iff.rfl | lemma | continuous_linear_map.is_self_adjoint_iff' | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"is_self_adjoint"
] | A continuous linear operator is self-adjoint iff it is equal to its adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adjoint_pair_inner (A : E' →L[ℝ] F') :
linear_map.is_adjoint_pair (sesq_form_of_inner : E' →ₗ[ℝ] E' →ₗ[ℝ] ℝ)
(sesq_form_of_inner : F' →ₗ[ℝ] F' →ₗ[ℝ] ℝ) A (A†) | λ x y, by simp only [sesq_form_of_inner_apply_apply, adjoint_inner_left, to_linear_map_eq_coe,
coe_coe] | lemma | continuous_linear_map.is_adjoint_pair_inner | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"coe_coe",
"linear_map.is_adjoint_pair",
"sesq_form_of_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_eq {A : E →L[𝕜] E} (hA : is_self_adjoint A) : A.adjoint = A | hA | lemma | is_self_adjoint.adjoint_eq | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric {A : E →L[𝕜] E} (hA : is_self_adjoint A) :
(A : E →ₗ[𝕜] E).is_symmetric | λ x y, by rw_mod_cast [←A.adjoint_inner_right, hA.adjoint_eq] | lemma | is_self_adjoint.is_symmetric | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"is_self_adjoint"
] | Every self-adjoint operator on an inner product space is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_adjoint {T : E →L[𝕜] E} (hT : is_self_adjoint T) (S : E →L[𝕜] F) :
is_self_adjoint (S ∘L T ∘L S.adjoint) | begin
rw is_self_adjoint_iff' at ⊢ hT,
simp only [hT, adjoint_comp, adjoint_adjoint],
exact continuous_linear_map.comp_assoc _ _ _,
end | lemma | is_self_adjoint.conj_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.comp_assoc",
"is_self_adjoint"
] | Conjugating preserves self-adjointness | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_conj {T : E →L[𝕜] E} (hT : is_self_adjoint T) (S : F →L[𝕜] E) :
is_self_adjoint (S.adjoint ∘L T ∘L S) | begin
rw is_self_adjoint_iff' at ⊢ hT,
simp only [hT, adjoint_comp, adjoint_adjoint],
exact continuous_linear_map.comp_assoc _ _ _,
end | lemma | is_self_adjoint.adjoint_conj | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.comp_assoc",
"is_self_adjoint"
] | Conjugating preserves self-adjointness | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_map.is_self_adjoint_iff_is_symmetric {A : E →L[𝕜] E} :
is_self_adjoint A ↔ (A : E →ₗ[𝕜] E).is_symmetric | ⟨λ hA, hA.is_symmetric, λ hA, ext $ λ x, ext_inner_right 𝕜 $
λ y, (A.adjoint_inner_left y x).symm ▸ (hA x y).symm⟩ | lemma | continuous_linear_map.is_self_adjoint_iff_is_symmetric | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"ext_inner_right",
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_map.is_symmetric.is_self_adjoint {A : E →L[𝕜] E}
(hA : (A : E →ₗ[𝕜] E).is_symmetric) : is_self_adjoint A | by rwa ←continuous_linear_map.is_self_adjoint_iff_is_symmetric at hA | lemma | linear_map.is_symmetric.is_self_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.orthogonal_projection_is_self_adjoint (U : submodule 𝕜 E)
[complete_space U] :
is_self_adjoint (U.subtypeL ∘L orthogonal_projection U) | (orthogonal_projection_is_symmetric U).is_self_adjoint | lemma | orthogonal_projection_is_self_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"is_self_adjoint",
"orthogonal_projection",
"orthogonal_projection_is_symmetric",
"submodule"
] | The orthogonal projection is self-adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_orthogonal_projection {T : E →L[𝕜] E}
(hT : is_self_adjoint T) (U : submodule 𝕜 E) [complete_space U] :
is_self_adjoint (U.subtypeL ∘L orthogonal_projection U ∘L T ∘L U.subtypeL ∘L
orthogonal_projection U) | begin
rw ←continuous_linear_map.comp_assoc,
nth_rewrite 0 ←(orthogonal_projection_is_self_adjoint U).adjoint_eq,
refine hT.adjoint_conj _,
end | lemma | is_self_adjoint.conj_orthogonal_projection | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"is_self_adjoint",
"orthogonal_projection",
"orthogonal_projection_is_self_adjoint",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric.to_self_adjoint (hT : is_symmetric T) : self_adjoint (E →L[𝕜] E) | ⟨⟨T, hT.continuous⟩, continuous_linear_map.is_self_adjoint_iff_is_symmetric.mpr hT⟩ | def | linear_map.is_symmetric.to_self_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"self_adjoint"
] | The **Hellinger--Toeplitz theorem**: Construct a self-adjoint operator from an everywhere
defined symmetric operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.coe_to_self_adjoint (hT : is_symmetric T) :
(hT.to_self_adjoint : E →ₗ[𝕜] E) = T | rfl | lemma | linear_map.is_symmetric.coe_to_self_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric.to_self_adjoint_apply (hT : is_symmetric T) {x : E} :
hT.to_self_adjoint x = T x | rfl | lemma | linear_map.is_symmetric.to_self_adjoint_apply | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint : (E →ₗ[𝕜] F) ≃ₗ⋆[𝕜] (F →ₗ[𝕜] E) | ((linear_map.to_continuous_linear_map : (E →ₗ[𝕜] F) ≃ₗ[𝕜] (E →L[𝕜] F)).trans
continuous_linear_map.adjoint.to_linear_equiv).trans
linear_map.to_continuous_linear_map.symm | def | linear_map.adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"linear_map.to_continuous_linear_map"
] | The adjoint of an operator from the finite-dimensional inner product space E to the finite-
dimensional inner product space F. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_to_continuous_linear_map (A : E →ₗ[𝕜] F) :
A.adjoint.to_continuous_linear_map = A.to_continuous_linear_map.adjoint | rfl | lemma | linear_map.adjoint_to_continuous_linear_map | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_eq_to_clm_adjoint (A : E →ₗ[𝕜] F) :
A.adjoint = A.to_continuous_linear_map.adjoint | rfl | lemma | linear_map.adjoint_eq_to_clm_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_inner_left (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪adjoint A y, x⟫ = ⟪y, A x⟫ | begin
rw [←coe_to_continuous_linear_map A, adjoint_eq_to_clm_adjoint],
exact continuous_linear_map.adjoint_inner_left _ x y,
end | lemma | linear_map.adjoint_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.adjoint_inner_left"
] | The fundamental property of the adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_inner_right (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪x, adjoint A y⟫ = ⟪A x, y⟫ | begin
rw [←coe_to_continuous_linear_map A, adjoint_eq_to_clm_adjoint],
exact continuous_linear_map.adjoint_inner_right _ x y,
end | lemma | linear_map.adjoint_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_map.adjoint_inner_right"
] | The fundamental property of the adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_adjoint (A : E →ₗ[𝕜] F) : A.adjoint.adjoint = A | begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
rw [adjoint_inner_right, adjoint_inner_left],
end | lemma | linear_map.adjoint_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"ext_inner_left"
] | The adjoint is involutive | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_comp (A : F →ₗ[𝕜] G) (B : E →ₗ[𝕜] F) :
(A ∘ₗ B).adjoint = B.adjoint ∘ₗ A.adjoint | begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
simp only [adjoint_inner_right, linear_map.coe_comp, function.comp_app],
end | lemma | linear_map.adjoint_comp | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"ext_inner_left",
"linear_map.coe_comp"
] | The adjoint of the composition of two operators is the composition of the two adjoints
in reverse order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_adjoint_iff (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫) | begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
ext x,
exact ext_inner_right 𝕜 (λ y, by simp only [adjoint_inner_left, h x y])
end | lemma | linear_map.eq_adjoint_iff | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"ext_inner_right"
] | The adjoint is unique: a map `A` is the adjoint of `B` iff it satisfies `⟪A x, y⟫ = ⟪x, B y⟫`
for all `x` and `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_adjoint_iff_basis {ι₁ : Type*} {ι₂ : Type*} (b₁ : basis ι₁ 𝕜 E) (b₂ : basis ι₂ 𝕜 F)
(A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ (i₁ : ι₁) (i₂ : ι₂), ⟪A (b₁ i₁), b₂ i₂⟫ = ⟪b₁ i₁, B (b₂ i₂)⟫) | begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
refine basis.ext b₁ (λ i₁, _),
exact ext_inner_right_basis b₂ (λ i₂, by simp only [adjoint_inner_left, h i₁ i₂]),
end | lemma | linear_map.eq_adjoint_iff_basis | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"basis",
"basis.ext"
] | The adjoint is unique: a map `A` is the adjoint of `B` iff it satisfies `⟪A x, y⟫ = ⟪x, B y⟫`
for all basis vectors `x` and `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_adjoint_iff_basis_left {ι : Type*} (b : basis ι 𝕜 E) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ i y, ⟪A (b i), y⟫ = ⟪b i, B y⟫) | begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, basis.ext b (λ i, _)⟩,
exact ext_inner_right 𝕜 (λ y, by simp only [h i, adjoint_inner_left]),
end | lemma | linear_map.eq_adjoint_iff_basis_left | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"basis",
"basis.ext",
"ext_inner_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_adjoint_iff_basis_right {ι : Type*} (b : basis ι 𝕜 F) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ i x, ⟪A x, b i⟫ = ⟪x, B (b i)⟫) | begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
ext x,
refine ext_inner_right_basis b (λ i, by simp only [h i, adjoint_inner_left]),
end | lemma | linear_map.eq_adjoint_iff_basis_right | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_adjoint (A : E →ₗ[𝕜] E) : star A = A.adjoint | rfl | lemma | linear_map.star_eq_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint_iff' {A : E →ₗ[𝕜] E} : is_self_adjoint A ↔ A.adjoint = A | iff.rfl | lemma | linear_map.is_self_adjoint_iff' | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"is_self_adjoint"
] | A continuous linear operator is self-adjoint iff it is equal to its adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric_iff_is_self_adjoint (A : E →ₗ[𝕜] E) :
is_symmetric A ↔ is_self_adjoint A | by { rw [is_self_adjoint_iff', is_symmetric, ← linear_map.eq_adjoint_iff], exact eq_comm } | lemma | linear_map.is_symmetric_iff_is_self_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"is_self_adjoint",
"linear_map.eq_adjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_adjoint_pair_inner (A : E' →ₗ[ℝ] F') :
is_adjoint_pair (sesq_form_of_inner : E' →ₗ[ℝ] E' →ₗ[ℝ] ℝ)
(sesq_form_of_inner : F' →ₗ[ℝ] F' →ₗ[ℝ] ℝ) A A.adjoint | λ x y, by simp only [sesq_form_of_inner_apply_apply, adjoint_inner_left] | lemma | linear_map.is_adjoint_pair_inner | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"sesq_form_of_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric_adjoint_mul_self (T : E →ₗ[𝕜] E) : is_symmetric (T.adjoint * T) | λ x y, by simp only [mul_apply, adjoint_inner_left, adjoint_inner_right] | lemma | linear_map.is_symmetric_adjoint_mul_self | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [] | The Gram operator T†T is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_inner_adjoint_mul_self_nonneg (T : E →ₗ[𝕜] E) (x : E) :
0 ≤ re ⟪ x, (T.adjoint * T) x ⟫ | by {simp only [mul_apply, adjoint_inner_right,
inner_self_eq_norm_sq_to_K], norm_cast, exact sq_nonneg _} | lemma | linear_map.re_inner_adjoint_mul_self_nonneg | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"inner_self_eq_norm_sq_to_K",
"sq_nonneg"
] | The Gram operator T†T is a positive operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_inner_adjoint_mul_self_eq_zero (T : E →ₗ[𝕜] E) (x : E) :
im ⟪ x, linear_map.adjoint T (T x) ⟫ = 0 | by {simp only [mul_apply,
adjoint_inner_right, inner_self_eq_norm_sq_to_K], norm_cast} | lemma | linear_map.im_inner_adjoint_mul_self_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"inner_self_eq_norm_sq_to_K",
"linear_map.adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_euclidean_lin_conj_transpose_eq_adjoint (A : matrix m n 𝕜) :
A.conj_transpose.to_euclidean_lin = A.to_euclidean_lin.adjoint | begin
rw linear_map.eq_adjoint_iff,
intros x y,
simp_rw [euclidean_space.inner_eq_star_dot_product, pi_Lp_equiv_to_euclidean_lin,
to_lin'_apply, star_mul_vec, conj_transpose_conj_transpose, dot_product_mul_vec],
end | lemma | matrix.to_euclidean_lin_conj_transpose_eq_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/adjoint.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.pi_L2"
] | [
"euclidean_space.inner_eq_star_dot_product",
"linear_map.eq_adjoint_iff",
"matrix"
] | The adjoint of the linear map associated to a matrix is the linear map associated to the
conjugate transpose of that matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_inner (𝕜 E : Type*) | (inner : E → E → 𝕜) | class | has_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | Syntactic typeclass for types endowed with an inner product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space (𝕜 : Type*) (E : Type*) [is_R_or_C 𝕜] [normed_add_comm_group E]
extends normed_space 𝕜 E, has_inner 𝕜 E | (norm_sq_eq_inner : ∀ (x : E), ‖x‖^2 = re (inner x x))
(conj_symm : ∀ x y, conj (inner y x) = inner x y)
(add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z)
(smul_left : ∀ x y r, inner (r • x) y = (conj r) * inner x y) | class | inner_product_space | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_inner",
"is_R_or_C",
"normed_add_comm_group",
"normed_space"
] | An inner product space is a vector space with an additional operation called inner product.
The norm could be derived from the inner product, instead we require the existence of a norm and
the fact that `‖x‖^2 = re ⟪x, x⟫` to be able to put instances on `𝕂` or product
spaces.
To construct a norm from an inner product... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space.core (𝕜 : Type*) (F : Type*)
[is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] extends has_inner 𝕜 F | (conj_symm : ∀ x y, conj (inner y x) = inner x y)
(nonneg_re : ∀ x, 0 ≤ re (inner x x))
(definite : ∀ x, inner x x = 0 → x = 0)
(add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z)
(smul_left : ∀ x y r, inner (r • x) y = (conj r) * inner x y) | structure | inner_product_space.core | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"add_comm_group",
"has_inner",
"is_R_or_C",
"module"
] | A structure requiring that a scalar product is positive definite and symmetric, from which one
can construct an `inner_product_space` instance in `inner_product_space.of_core`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space.to_core [normed_add_comm_group E] [c : inner_product_space 𝕜 E] :
inner_product_space.core 𝕜 E | { nonneg_re := λ x, by { rw [← inner_product_space.norm_sq_eq_inner], apply sq_nonneg },
definite := λ x hx, norm_eq_zero.1 $ pow_eq_zero $
by rw [inner_product_space.norm_sq_eq_inner x, hx, map_zero],
.. c } | def | inner_product_space.to_core | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_product_space",
"inner_product_space.core",
"normed_add_comm_group",
"pow_eq_zero",
"sq_nonneg"
] | Define `inner_product_space.core` from `inner_product_space`. Defined to reuse lemmas about
`inner_product_space.core` for `inner_product_space`s. Note that the `has_norm` instance provided by
`inner_product_space.core.has_norm` is propositionally but not definitionally equal to the original
norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_has_inner' : has_inner 𝕜 F | c.to_has_inner | def | inner_product_space.core.to_has_inner' | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_inner"
] | Inner product defined by the `inner_product_space.core` structure. We can't reuse
`inner_product_space.core.to_has_inner` because it takes `inner_product_space.core` as an explicit
argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sq (x : F) | reK ⟪x, x⟫ | def | inner_product_space.core.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | The norm squared function for `inner_product_space.core` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_conj_symm (x y : F) : ⟪y, x⟫† = ⟪x, y⟫ | c.conj_symm x y | lemma | inner_product_space.core.inner_conj_symm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_conj_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_nonneg {x : F} : 0 ≤ re ⟪x, x⟫ | c.nonneg_re _ | lemma | inner_product_space.core.inner_self_nonneg | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_im (x : F) : im ⟪x, x⟫ = 0 | by rw [← @of_real_inj 𝕜, im_eq_conj_sub]; simp [inner_conj_symm] | lemma | inner_product_space.core.inner_self_im | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_conj_symm",
"inner_self_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_add_left (x y z : F) : ⟪x + y, z⟫ = ⟪x, z⟫ + ⟪y, z⟫ | c.add_left _ _ _ | lemma | inner_product_space.core.inner_add_left | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_add_right (x y z : F) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ | by rw [←inner_conj_symm, inner_add_left, ring_hom.map_add]; simp only [inner_conj_symm] | lemma | inner_product_space.core.inner_add_right | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_left",
"inner_add_right",
"inner_conj_symm",
"ring_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_norm_sq_eq_inner_self (x : F) : (norm_sqF x : 𝕜) = ⟪x, x⟫ | begin
rw ext_iff,
exact ⟨by simp only [of_real_re]; refl, by simp only [inner_self_im, of_real_im]⟩
end | lemma | inner_product_space.core.coe_norm_sq_eq_inner_self | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_re_symm (x y : F) : re ⟪x, y⟫ = re ⟪y, x⟫ | by rw [←inner_conj_symm, conj_re] | lemma | inner_product_space.core.inner_re_symm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_re_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_im_symm (x y : F) : im ⟪x, y⟫ = -im ⟪y, x⟫ | by rw [←inner_conj_symm, conj_im] | lemma | inner_product_space.core.inner_im_symm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_im_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_smul_left (x y : F) {r : 𝕜} : ⟪r • x, y⟫ = r† * ⟪x, y⟫ | c.smul_left _ _ _ | lemma | inner_product_space.core.inner_smul_left | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫ | by rw [←inner_conj_symm, inner_smul_left]; simp only [conj_conj, inner_conj_symm, ring_hom.map_mul] | lemma | inner_product_space.core.inner_smul_right | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_conj_symm",
"inner_smul_left",
"inner_smul_right",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_zero_left (x : F) : ⟪0, x⟫ = 0 | by rw [←zero_smul 𝕜 (0 : F), inner_smul_left]; simp only [zero_mul, ring_hom.map_zero] | lemma | inner_product_space.core.inner_zero_left | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_smul_left",
"inner_zero_left",
"ring_hom.map_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_zero_right (x : F) : ⟪x, 0⟫ = 0 | by rw [←inner_conj_symm, inner_zero_left]; simp only [ring_hom.map_zero] | lemma | inner_product_space.core.inner_zero_right | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_zero_left",
"inner_zero_right",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_eq_zero {x : F} : ⟪x, x⟫ = 0 ↔ x = 0 | ⟨c.definite _, by { rintro rfl, exact inner_zero_left _ }⟩ | lemma | inner_product_space.core.inner_self_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_eq_zero",
"inner_zero_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_eq_zero {x : F} : norm_sqF x = 0 ↔ x = 0 | iff.trans (by simp only [norm_sq, ext_iff, map_zero, inner_self_im, eq_self_iff_true, and_true])
(@inner_self_eq_zero 𝕜 _ _ _ _ _ x) | lemma | inner_product_space.core.norm_sq_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_eq_zero",
"inner_self_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_ne_zero {x : F} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 | inner_self_eq_zero.not | lemma | inner_product_space.core.inner_self_ne_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_re_to_K (x : F) : (re ⟪x, x⟫ : 𝕜) = ⟪x, x⟫ | by norm_num [ext_iff, inner_self_im] | lemma | inner_product_space.core.inner_self_re_to_K | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_im",
"inner_self_re_to_K"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inner_symm (x y : F) : ‖⟪x, y⟫‖ = ‖⟪y, x⟫‖ | by rw [←inner_conj_symm, norm_conj] | lemma | inner_product_space.core.norm_inner_symm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_inner_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_neg_left (x y : F) : ⟪-x, y⟫ = -⟪x, y⟫ | by { rw [← neg_one_smul 𝕜 x, inner_smul_left], simp } | lemma | inner_product_space.core.inner_neg_left | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_neg_left",
"inner_smul_left",
"neg_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_neg_right (x y : F) : ⟪x, -y⟫ = -⟪x, y⟫ | by rw [←inner_conj_symm, inner_neg_left]; simp only [ring_hom.map_neg, inner_conj_symm] | lemma | inner_product_space.core.inner_neg_right | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_conj_symm",
"inner_neg_left",
"inner_neg_right",
"ring_hom.map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_sub_left (x y z : F) : ⟪x - y, z⟫ = ⟪x, z⟫ - ⟪y, z⟫ | by { simp [sub_eq_add_neg, inner_add_left, inner_neg_left] } | lemma | inner_product_space.core.inner_sub_left | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_left",
"inner_neg_left",
"inner_sub_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_sub_right (x y z : F) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ | by { simp [sub_eq_add_neg, inner_add_right, inner_neg_right] } | lemma | inner_product_space.core.inner_sub_right | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_right",
"inner_neg_right",
"inner_sub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_mul_symm_re_eq_norm (x y : F) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ | by { rw [←inner_conj_symm, mul_comm], exact re_eq_norm_of_mul_conj (inner y x), } | lemma | inner_product_space.core.inner_mul_symm_re_eq_norm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_mul_symm_re_eq_norm",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_add_add_self (x y : F) : ⟪x + y, x + y⟫ = ⟪x, x⟫ + ⟪x, y⟫ + ⟪y, x⟫ + ⟪y, y⟫ | by simp only [inner_add_left, inner_add_right]; ring | lemma | inner_product_space.core.inner_add_add_self | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_add_self",
"inner_add_left",
"inner_add_right",
"ring"
] | Expand `inner (x + y) (x + y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ | by simp only [inner_sub_left, inner_sub_right]; ring | lemma | inner_product_space.core.inner_sub_sub_self | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_sub_left",
"inner_sub_right",
"inner_sub_sub_self",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_schwarz_aux (x y : F) :
norm_sqF (⟪x, y⟫ • x - ⟪x, x⟫ • y) =
norm_sqF x * (norm_sqF x * norm_sqF y - ‖⟪x, y⟫‖ ^ 2) | begin
rw [← @of_real_inj 𝕜, coe_norm_sq_eq_inner_self],
simp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_of_real, mul_sub,
← coe_norm_sq_eq_inner_self x, ← coe_norm_sq_eq_inner_self y],
rw [← mul_assoc, mul_conj, is_R_or_C.conj_mul, norm_sq_eq_def',
mul_left_comm, ← inner_conj_symm ... | theorem | inner_product_space.core.cauchy_schwarz_aux | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_conj_symm",
"inner_smul_left",
"inner_smul_right",
"inner_sub_sub_self",
"is_R_or_C.conj_mul",
"mul_assoc",
"mul_left_comm",
"ring"
] | An auxiliary equality useful to prove the **Cauchy–Schwarz inequality**: the square of the norm
of `⟪x, y⟫ • x - ⟪x, x⟫ • y` is equal to `‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2 - ‖⟪x, y⟫‖ ^ 2)`. We use
`inner_product_space.of_core.norm_sq x` etc (defeq to `is_R_or_C.re ⟪x, x⟫`) instead of `‖x‖ ^ 2`
etc to avoid extra rewrites wh... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_mul_inner_self_le (x y : F) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ | begin
rcases eq_or_ne x 0 with (rfl | hx),
{ simp only [inner_zero_left, map_zero, zero_mul, norm_zero] },
{ have hx' : 0 < norm_sqF x := inner_self_nonneg.lt_of_ne' (mt norm_sq_eq_zero.1 hx),
rw [← sub_nonneg, ← mul_nonneg_iff_right_nonneg_of_pos hx', ← norm_sq, ← norm_sq,
norm_inner_symm y, ← sq, ← ca... | lemma | inner_product_space.core.inner_mul_inner_self_le | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"eq_or_ne",
"inner_mul_inner_self_le",
"inner_self_nonneg",
"inner_zero_left",
"mul_nonneg_iff_right_nonneg_of_pos",
"norm_inner_symm",
"zero_mul"
] | **Cauchy–Schwarz inequality**.
We need this for the `core` structure to prove the triangle inequality below when
showing the core is a normed group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_has_norm : has_norm F | { norm := λ x, sqrt (re ⟪x, x⟫) } | def | inner_product_space.core.to_has_norm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_norm"
] | Norm constructed from a `inner_product_space.core` structure, defined to be the square root
of the scalar product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_sqrt_inner (x : F) : ‖x‖ = sqrt (re ⟪x, x⟫) | rfl | lemma | inner_product_space.core.norm_eq_sqrt_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_eq_sqrt_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_eq_norm_mul_norm (x : F) : re ⟪x, x⟫ = ‖x‖ * ‖x‖ | by rw [norm_eq_sqrt_inner, ←sqrt_mul inner_self_nonneg (re ⟪x, x⟫),
sqrt_mul_self inner_self_nonneg] | lemma | inner_product_space.core.inner_self_eq_norm_mul_norm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_eq_norm_mul_norm",
"inner_self_nonneg",
"norm_eq_sqrt_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_norm_sq_eq_norm (x : F) : sqrt (norm_sqF x) = ‖x‖ | rfl | lemma | inner_product_space.core.sqrt_norm_sq_eq_norm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inner_le_norm (x y : F) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ | nonneg_le_nonneg_of_sq_le_sq (mul_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) $
calc ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ = ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ : by rw [norm_inner_symm]
... ≤ re ⟪x, x⟫ * re ⟪y, y⟫ : inner_mul_inner_self_le x y
... = ‖x‖ * ‖y‖ * (‖x‖ * ‖y‖) : by simp only [inner_self_eq_norm_mul_norm]; ring | lemma | inner_product_space.core.norm_inner_le_norm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_mul_inner_self_le",
"inner_self_eq_norm_mul_norm",
"nonneg_le_nonneg_of_sq_le_sq",
"norm_inner_le_norm",
"norm_inner_symm",
"ring"
] | Cauchy–Schwarz inequality with norm | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_add_comm_group : normed_add_comm_group F | add_group_norm.to_normed_add_comm_group
{ to_fun := λ x, sqrt (re ⟪x, x⟫),
map_zero' := by simp only [sqrt_zero, inner_zero_right, map_zero],
neg' := λ x, by simp only [inner_neg_left, neg_neg, inner_neg_right],
add_le' := λ x y, begin
have h₁ : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm _ _,
have h₂ : re... | def | inner_product_space.core.to_normed_add_comm_group | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_add_self",
"inner_neg_left",
"inner_neg_right",
"inner_self_nonneg",
"inner_zero_right",
"mul_comm",
"nonneg_le_nonneg_of_sq_le_sq",
"norm_inner_le_norm",
"normed_add_comm_group"
] | Normed group structure constructed from an `inner_product_space.core` structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_space : normed_space 𝕜 F | { norm_smul_le := assume r x,
begin
rw [norm_eq_sqrt_inner, inner_smul_left, inner_smul_right, ←mul_assoc],
rw [is_R_or_C.conj_mul, of_real_mul_re, sqrt_mul, ← coe_norm_sq_eq_inner_self, of_real_re],
{ simp [sqrt_norm_sq_eq_norm, is_R_or_C.sqrt_norm_sq_eq_norm] },
{ exact norm_sq_nonneg r }
end } | def | inner_product_space.core.to_normed_space | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_smul_left",
"inner_smul_right",
"is_R_or_C.conj_mul",
"is_R_or_C.sqrt_norm_sq_eq_norm",
"norm_eq_sqrt_inner",
"norm_smul_le",
"normed_space"
] | Normed space structure constructed from a `inner_product_space.core` structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space.of_core [add_comm_group F] [module 𝕜 F]
(c : inner_product_space.core 𝕜 F) : inner_product_space 𝕜 F | begin
letI : normed_space 𝕜 F := @inner_product_space.core.to_normed_space 𝕜 F _ _ _ c,
exact { norm_sq_eq_inner := λ x,
begin
have h₁ : ‖x‖^2 = (sqrt (re (c.inner x x))) ^ 2 := rfl,
have h₂ : 0 ≤ re (c.inner x x) := inner_product_space.core.inner_self_nonneg,
simp [h₁, sq_sqrt, h₂],
end... | def | inner_product_space.of_core | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"add_comm_group",
"inner_product_space",
"inner_product_space.core",
"inner_product_space.core.inner_self_nonneg",
"inner_product_space.core.to_normed_space",
"module",
"normed_space"
] | Given a `inner_product_space.core` structure on a space, one can use it to turn
the space into an inner product space. The `normed_add_comm_group` structure is expected
to already be defined with `inner_product_space.of_core.to_normed_add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_conj_symm (x y : E) : ⟪y, x⟫† = ⟪x, y⟫ | inner_product_space.conj_symm _ _ | lemma | inner_conj_symm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_comm (x y : F) : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ | @inner_conj_symm ℝ _ _ _ _ x y | lemma | real_inner_comm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_conj_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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