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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_max_on.norm_add_self (h : is_max_on (norm ∘ f) s c) : is_max_on (λ x, ‖f x + f c‖) s c | h.norm_add_self | lemma | is_max_on.norm_add_self | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_max_on"
] | If `f : α → E` is a function such that `norm ∘ f` has a maximum on a set `s` at a point `c`,
then the function `λ x, ‖f x + f c‖` has a maximul on `s` at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max_on.norm_add_same_ray (h : is_local_max_on (norm ∘ f) s c)
(hy : same_ray ℝ (f c) y) : is_local_max_on (λ x, ‖f x + y‖) s c | h.norm_add_same_ray hy | lemma | is_local_max_on.norm_add_same_ray | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_local_max_on",
"same_ray"
] | If `f : α → E` is a function such that `norm ∘ f` has a local maximum on a set `s` at a point
`c` and `y` is a vector on the same ray as `f c`, then the function `λ x, ‖f x + y‖` has a local
maximul on `s` at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max_on.norm_add_self (h : is_local_max_on (norm ∘ f) s c) :
is_local_max_on (λ x, ‖f x + f c‖) s c | h.norm_add_self | lemma | is_local_max_on.norm_add_self | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_local_max_on"
] | If `f : α → E` is a function such that `norm ∘ f` has a local maximum on a set `s` at a point
`c`, then the function `λ x, ‖f x + f c‖` has a local maximul on `s` at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max.norm_add_same_ray (h : is_local_max (norm ∘ f) c)
(hy : same_ray ℝ (f c) y) : is_local_max (λ x, ‖f x + y‖) c | h.norm_add_same_ray hy | lemma | is_local_max.norm_add_same_ray | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_local_max",
"same_ray"
] | If `f : α → E` is a function such that `norm ∘ f` has a local maximum at a point `c` and `y` is
a vector on the same ray as `f c`, then the function `λ x, ‖f x + y‖` has a local maximul at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max.norm_add_self (h : is_local_max (norm ∘ f) c) :
is_local_max (λ x, ‖f x + f c‖) c | h.norm_add_self | lemma | is_local_max.norm_add_self | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_local_max"
] | If `f : α → E` is a function such that `norm ∘ f` has a local maximum at a point `c`, then the
function `λ x, ‖f x + f c‖` has a local maximul at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_isometry_equiv
(li : E₁ →ₗᵢ[R₁] F) (h : finrank R₁ E₁ = finrank R₁ F) : E₁ ≃ₗᵢ[R₁] F | { to_linear_equiv :=
li.to_linear_map.linear_equiv_of_injective li.injective h,
norm_map' := li.norm_map' } | def | linear_isometry.to_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | A linear isometry between finite dimensional spaces of equal dimension can be upgraded
to a linear isometry equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_linear_isometry_equiv
(li : E₁ →ₗᵢ[R₁] F) (h : finrank R₁ E₁ = finrank R₁ F) :
(li.to_linear_isometry_equiv h : E₁ → F) = li | rfl | lemma | linear_isometry.coe_to_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_isometry_equiv_apply
(li : E₁ →ₗᵢ[R₁] F) (h : finrank R₁ E₁ = finrank R₁ F) (x : E₁) :
(li.to_linear_isometry_equiv h) x = li x | rfl | lemma | linear_isometry.to_linear_isometry_equiv_apply | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_equiv [inhabited P₁]
(li : P₁ →ᵃⁱ[𝕜] P₂) (h : finrank 𝕜 V₁ = finrank 𝕜 V₂) : P₁ ≃ᵃⁱ[𝕜] P₂ | affine_isometry_equiv.mk' li (li.linear_isometry.to_linear_isometry_equiv h) (arbitrary P₁)
(λ p, by simp) | def | affine_isometry.to_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"affine_isometry_equiv.mk'"
] | An affine isometry between finite dimensional spaces of equal dimension can be upgraded
to an affine isometry equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_affine_isometry_equiv [inhabited P₁]
(li : P₁ →ᵃⁱ[𝕜] P₂) (h : finrank 𝕜 V₁ = finrank 𝕜 V₂) :
(li.to_affine_isometry_equiv h : P₁ → P₂) = li | rfl | lemma | affine_isometry.coe_to_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_equiv_apply [inhabited P₁]
(li : P₁ →ᵃⁱ[𝕜] P₂) (h : finrank 𝕜 V₁ = finrank 𝕜 V₂) (x : P₁) :
(li.to_affine_isometry_equiv h) x = li x | rfl | lemma | affine_isometry.to_affine_isometry_equiv_apply | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_map.continuous_of_finite_dimensional (f : PE →ᵃ[𝕜] PF) : continuous f | affine_map.continuous_linear_iff.1 f.linear.continuous_of_finite_dimensional | theorem | affine_map.continuous_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_equiv.continuous_of_finite_dimensional (f : PE ≃ᵃ[𝕜] PF) : continuous f | f.to_affine_map.continuous_of_finite_dimensional | theorem | affine_equiv.continuous_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_equiv.to_homeomorph_of_finite_dimensional (f : PE ≃ᵃ[𝕜] PF) : PE ≃ₜ PF | { to_equiv := f.to_equiv,
continuous_to_fun := f.continuous_of_finite_dimensional,
continuous_inv_fun :=
begin
haveI : finite_dimensional 𝕜 F, from f.linear.finite_dimensional,
exact f.symm.continuous_of_finite_dimensional
end } | def | affine_equiv.to_homeomorph_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional"
] | Reinterpret an affine equivalence as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_equiv.coe_to_homeomorph_of_finite_dimensional (f : PE ≃ᵃ[𝕜] PF) :
⇑f.to_homeomorph_of_finite_dimensional = f | rfl | lemma | affine_equiv.coe_to_homeomorph_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_equiv.coe_to_homeomorph_of_finite_dimensional_symm (f : PE ≃ᵃ[𝕜] PF) :
⇑f.to_homeomorph_of_finite_dimensional.symm = f.symm | rfl | lemma | affine_equiv.coe_to_homeomorph_of_finite_dimensional_symm | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.continuous_det :
continuous (λ (f : E →L[𝕜] E), f.det) | begin
change continuous (λ (f : E →L[𝕜] E), (f : E →ₗ[𝕜] E).det),
by_cases h : ∃ (s : finset E), nonempty (basis ↥s 𝕜 E),
{ rcases h with ⟨s, ⟨b⟩⟩,
haveI : finite_dimensional 𝕜 E := finite_dimensional.of_fintype_basis b,
simp_rw linear_map.det_eq_det_to_matrix_of_finset b,
refine continuous.matrix... | lemma | continuous_linear_map.continuous_det | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis",
"continuous",
"continuous.matrix_det",
"continuous_const",
"continuous_linear_map.coe_lm",
"finite_dimensional",
"finite_dimensional.of_fintype_basis",
"finset",
"linear_map.det",
"linear_map.det_eq_det_to_matrix_of_finset",
"linear_map.to_matrix",
"monoid_hom.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_extension_constant
(E' : Type*) [normed_add_comm_group E'] [normed_space ℝ E'] [finite_dimensional ℝ E'] : ℝ≥0 | let A := (basis.of_vector_space ℝ E').equiv_fun.to_continuous_linear_equiv in
max (‖A.symm.to_continuous_linear_map‖₊ * ‖A.to_continuous_linear_map‖₊) 1 | def | lipschitz_extension_constant | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis.of_vector_space",
"finite_dimensional",
"normed_add_comm_group",
"normed_space"
] | Any `K`-Lipschitz map from a subset `s` of a metric space `α` to a finite-dimensional real
vector space `E'` can be extended to a Lipschitz map on the whole space `α`, with a slightly worse
constant `C * K` where `C` only depends on `E'`. We record a working value for this constant `C`
as `lipschitz_extension_constant ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_extension_constant_pos
(E' : Type*) [normed_add_comm_group E'] [normed_space ℝ E'] [finite_dimensional ℝ E'] :
0 < lipschitz_extension_constant E' | by { rw lipschitz_extension_constant, exact zero_lt_one.trans_le (le_max_right _ _) } | lemma | lipschitz_extension_constant_pos | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"lipschitz_extension_constant",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with.extend_finite_dimension
{α : Type*} [pseudo_metric_space α]
{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] [finite_dimensional ℝ E']
{s : set α} {f : α → E'} {K : ℝ≥0} (hf : lipschitz_on_with K f s) :
∃ (g : α → E'), lipschitz_with (lipschitz_extension_constant E' * K) g ∧ eq_on f... | begin
/- This result is already known for spaces `ι → ℝ`. We use a continuous linear equiv between
`E'` and such a space to transfer the result to `E'`. -/
let ι : Type* := basis.of_vector_space_index ℝ E',
let A := (basis.of_vector_space ℝ E').equiv_fun.to_continuous_linear_equiv,
have LA : lipschitz_with (‖... | theorem | lipschitz_on_with.extend_finite_dimension | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis.of_vector_space",
"basis.of_vector_space_index",
"finite_dimensional",
"le_rfl",
"lipschitz_extension_constant",
"lipschitz_on_with",
"lipschitz_with",
"mul_assoc",
"mul_le_mul'",
"normed_add_comm_group",
"normed_space",
"pseudo_metric_space"
] | Any `K`-Lipschitz map from a subset `s` of a metric space `α` to a finite-dimensional real
vector space `E'` can be extended to a Lipschitz map on the whole space `α`, with a slightly worse
constant `lipschitz_extension_constant E' * K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.exists_antilipschitz_with [finite_dimensional 𝕜 E] (f : E →ₗ[𝕜] F)
(hf : f.ker = ⊥) : ∃ K > 0, antilipschitz_with K f | begin
casesI subsingleton_or_nontrivial E,
{ exact ⟨1, zero_lt_one, antilipschitz_with.of_subsingleton⟩ },
{ rw linear_map.ker_eq_bot at hf,
let e : E ≃L[𝕜] f.range := (linear_equiv.of_injective f hf).to_continuous_linear_equiv,
exact ⟨_, e.nnnorm_symm_pos, e.antilipschitz⟩ }
end | lemma | linear_map.exists_antilipschitz_with | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"antilipschitz_with",
"finite_dimensional",
"linear_equiv.of_injective",
"linear_map.ker_eq_bot",
"subsingleton_or_nontrivial",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent.eventually {ι} [finite ι] {f : ι → E}
(hf : linear_independent 𝕜 f) : ∀ᶠ g in 𝓝 f, linear_independent 𝕜 g | begin
casesI nonempty_fintype ι,
simp only [fintype.linear_independent_iff'] at hf ⊢,
rcases linear_map.exists_antilipschitz_with _ hf with ⟨K, K0, hK⟩,
have : tendsto (λ g : ι → E, ∑ i, ‖g i - f i‖) (𝓝 f) (𝓝 $ ∑ i, ‖f i - f i‖),
from tendsto_finset_sum _ (λ i hi, tendsto.norm $
((continuous_apply i... | lemma | linear_independent.eventually | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous_apply",
"finite",
"finset.sum_mul",
"fintype.linear_independent_iff'",
"gt_mem_nhds",
"linear_independent",
"linear_map.comp_apply",
"linear_map.exists_antilipschitz_with",
"linear_map.id_apply",
"linear_map.ker_eq_bot",
"linear_map.proj_apply",
"linear_map.smul_right_apply",
"li... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_set_of_linear_independent {ι : Type*} [finite ι] :
is_open {f : ι → E | linear_independent 𝕜 f} | is_open_iff_mem_nhds.2 $ λ f, linear_independent.eventually | lemma | is_open_set_of_linear_independent | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite",
"is_open",
"linear_independent",
"linear_independent.eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_set_of_nat_le_rank (n : ℕ) : is_open {f : E →L[𝕜] F | ↑n ≤ (f : E →ₗ[𝕜] F).rank } | begin
simp only [linear_map.le_rank_iff_exists_linear_independent_finset, set_of_exists, ← exists_prop],
refine is_open_bUnion (λ t ht, _),
have : continuous (λ f : E →L[𝕜] F, (λ x : (t : set E), f x)),
from continuous_pi (λ x, (continuous_linear_map.apply 𝕜 F (x : E)).continuous),
exact is_open_set_of_li... | lemma | is_open_set_of_nat_le_rank | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous",
"continuous_linear_map.apply",
"continuous_pi",
"exists_prop",
"is_open",
"is_open_bUnion",
"linear_map.le_rank_iff_exists_linear_independent_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.op_nnnorm_le {ι : Type*} [fintype ι] (v : basis ι 𝕜 E) {u : E →L[𝕜] F} (M : ℝ≥0)
(hu : ∀ i, ‖u (v i)‖₊ ≤ M) :
‖u‖₊ ≤ fintype.card ι • ‖v.equiv_funL.to_continuous_linear_map‖₊ * M | u.op_nnnorm_le_bound _ $ λ e, begin
set φ := v.equiv_funL.to_continuous_linear_map,
calc
‖u e‖₊ = ‖u (∑ i, v.equiv_fun e i • v i)‖₊ : by rw [v.sum_equiv_fun]
... = ‖∑ i, (v.equiv_fun e i) • (u $ v i)‖₊ : by simp [u.map_sum, linear_map.map_smul]
... ≤ ∑ i, ‖(v.equiv_fun e i) • (u $ v i)‖₊ : nnnorm_sum_le... | lemma | basis.op_nnnorm_le | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis",
"fintype",
"fintype.card",
"linear_map.map_smul",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_right_comm",
"nnnorm_smul",
"smul_mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.op_norm_le {ι : Type*} [fintype ι] (v : basis ι 𝕜 E) {u : E →L[𝕜] F} {M : ℝ}
(hM : 0 ≤ M) (hu : ∀ i, ‖u (v i)‖ ≤ M) :
‖u‖ ≤ fintype.card ι • ‖v.equiv_funL.to_continuous_linear_map‖ * M | by simpa using nnreal.coe_le_coe.mpr (v.op_nnnorm_le ⟨M, hM⟩ hu) | lemma | basis.op_norm_le | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis",
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.exists_op_nnnorm_le {ι : Type*} [finite ι] (v : basis ι 𝕜 E) :
∃ C > (0 : ℝ≥0), ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ i, ‖u (v i)‖₊ ≤ M) → ‖u‖₊ ≤ C*M | by casesI nonempty_fintype ι; exact
⟨max (fintype.card ι • ‖v.equiv_funL.to_continuous_linear_map‖₊) 1,
zero_lt_one.trans_le (le_max_right _ _),
λ u M hu, (v.op_nnnorm_le M hu).trans $ mul_le_mul_of_nonneg_right (le_max_left _ _) (zero_le M)⟩ | lemma | basis.exists_op_nnnorm_le | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis",
"finite",
"fintype.card",
"mul_le_mul_of_nonneg_right",
"nonempty_fintype"
] | A weaker version of `basis.op_nnnorm_le` that abstracts away the value of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.exists_op_norm_le {ι : Type*} [finite ι] (v : basis ι 𝕜 E) :
∃ C > (0 : ℝ), ∀ {u : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ i, ‖u (v i)‖ ≤ M) → ‖u‖ ≤ C*M | let ⟨C, hC, h⟩ := v.exists_op_nnnorm_le in ⟨C, hC, λ u, subtype.forall'.mpr h⟩ | lemma | basis.exists_op_norm_le | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"basis",
"finite"
] | A weaker version of `basis.op_norm_le` that abstracts away the value of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_dimensional.complete [finite_dimensional 𝕜 E] : complete_space E | begin
set e := continuous_linear_equiv.of_finrank_eq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm,
have : uniform_embedding e.to_linear_equiv.to_equiv.symm := e.symm.uniform_embedding,
exact (complete_space_congr this).1 (by apply_instance)
end | lemma | finite_dimensional.complete | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"complete_space",
"complete_space_congr",
"continuous_linear_equiv.of_finrank_eq",
"finite_dimensional",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.complete_of_finite_dimensional (s : submodule 𝕜 E) [finite_dimensional 𝕜 s] :
is_complete (s : set E) | complete_space_coe_iff_is_complete.1 (finite_dimensional.complete 𝕜 s) | lemma | submodule.complete_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"finite_dimensional.complete",
"is_complete",
"submodule"
] | A finite-dimensional subspace is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.closed_of_finite_dimensional (s : submodule 𝕜 E) [finite_dimensional 𝕜 s] :
is_closed (s : set E) | s.complete_of_finite_dimensional.is_closed | lemma | submodule.closed_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"is_closed",
"submodule"
] | A finite-dimensional subspace is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_subspace.closed_of_finite_dimensional {P : Type*} [metric_space P]
[normed_add_torsor E P] (s : affine_subspace 𝕜 P) [finite_dimensional 𝕜 s.direction] :
is_closed (s : set P) | s.is_closed_direction_iff.mp s.direction.closed_of_finite_dimensional | lemma | affine_subspace.closed_of_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"affine_subspace",
"finite_dimensional",
"is_closed",
"metric_space",
"normed_add_torsor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
(h : ¬ (finite_dimensional 𝕜 E)) (s : finset E) :
∃ (x : E), ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ | begin
let F := submodule.span 𝕜 (s : set E),
haveI : finite_dimensional 𝕜 F := module.finite_def.2
((submodule.fg_top _).2 (submodule.fg_def.2 ⟨s, finset.finite_to_set _, rfl⟩)),
have Fclosed : is_closed (F : set E) := submodule.closed_of_finite_dimensional _,
have : ∃ x, x ∉ F,
{ contrapose! h,
hav... | theorem | exists_norm_le_le_norm_sub_of_finset | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"finset",
"finset.finite_to_set",
"is_closed",
"riesz_lemma_of_norm_lt",
"submodule",
"submodule.closed_of_finite_dimensional",
"submodule.fg_top",
"submodule.span",
"submodule.subset_span"
] | In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
(h : ¬ (finite_dimensional 𝕜 E)) :
∃ f : ℕ → E, (∀ n, ‖f n‖ ≤ R) ∧ (∀ m n, m ≠ n → 1 ≤ ‖f m - f n‖) | begin
haveI : is_symm E (λ (x y : E), 1 ≤ ‖x - y‖),
{ constructor,
assume x y hxy,
rw ← norm_neg,
simpa },
apply exists_seq_of_forall_finset_exists' (λ (x : E), ‖x‖ ≤ R) (λ (x : E) (y : E), 1 ≤ ‖x - y‖),
assume s hs,
exact exists_norm_le_le_norm_sub_of_finset hc hR h s,
end | theorem | exists_seq_norm_le_one_le_norm_sub' | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"exists_norm_le_le_norm_sub_of_finset",
"exists_seq_of_forall_finset_exists'",
"finite_dimensional"
] | In an infinite-dimensional normed space, there exists a sequence of points which are all
bounded by `R` and at distance at least `1`. For a version not assuming `c` and `R`, see
`exists_seq_norm_le_one_le_norm_sub`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_seq_norm_le_one_le_norm_sub (h : ¬ (finite_dimensional 𝕜 E)) :
∃ (R : ℝ) (f : ℕ → E), (1 < R) ∧ (∀ n, ‖f n‖ ≤ R) ∧ (∀ m n, m ≠ n → 1 ≤ ‖f m - f n‖) | begin
obtain ⟨c, hc⟩ : ∃ (c : 𝕜), 1 < ‖c‖ := normed_field.exists_one_lt_norm 𝕜,
have A : ‖c‖ < ‖c‖ + 1, by linarith,
rcases exists_seq_norm_le_one_le_norm_sub' hc A h with ⟨f, hf⟩,
exact ⟨‖c‖ + 1, f, hc.trans A, hf.1, hf.2⟩
end | theorem | exists_seq_norm_le_one_le_norm_sub | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"exists_seq_norm_le_one_le_norm_sub'",
"finite_dimensional",
"normed_field.exists_one_lt_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_dimensional_of_is_compact_closed_ball₀ {r : ℝ} (rpos : 0 < r)
(h : is_compact (metric.closed_ball (0 : E) r)) : finite_dimensional 𝕜 E | begin
by_contra hfin,
obtain ⟨R, f, Rgt, fle, lef⟩ :
∃ (R : ℝ) (f : ℕ → E), (1 < R) ∧ (∀ n, ‖f n‖ ≤ R) ∧ (∀ m n, m ≠ n → 1 ≤ ‖f m - f n‖) :=
exists_seq_norm_le_one_le_norm_sub hfin,
have rRpos : 0 < r / R := div_pos rpos (zero_lt_one.trans Rgt),
obtain ⟨c, hc⟩ : ∃ (c : 𝕜), 0 < ‖c‖ ∧ ‖c‖ < (r / R) := ... | theorem | finite_dimensional_of_is_compact_closed_ball₀ | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"by_contra",
"cauchy_seq",
"div_pos",
"exists_seq_norm_le_one_le_norm_sub",
"finite_dimensional",
"is_compact",
"metric.closed_ball",
"metric.mem_closed_ball",
"mul_le_mul",
"mul_le_mul_of_nonneg_left",
"mul_one",
"norm_smul",
"normed_field.exists_norm_lt",
"strict_mono"
] | **Riesz's theorem**: if a closed ball with center zero of positive radius is compact in a vector
space, then the space is finite-dimensional. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_dimensional_of_is_compact_closed_ball {r : ℝ} (rpos : 0 < r) {c : E}
(h : is_compact (metric.closed_ball c r)) : finite_dimensional 𝕜 E | begin
apply finite_dimensional_of_is_compact_closed_ball₀ 𝕜 rpos,
have : continuous (λ x, -c + x), from continuous_const.add continuous_id,
simpa using h.image this,
end | theorem | finite_dimensional_of_is_compact_closed_ball | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous",
"continuous_id",
"finite_dimensional",
"finite_dimensional_of_is_compact_closed_ball₀",
"is_compact",
"metric.closed_ball"
] | **Riesz's theorem**: if a closed ball of positive radius is compact in a vector space, then the
space is finite-dimensional. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_mul_support.eq_one_or_finite_dimensional {X : Type*}
[topological_space X] [has_one X] [t2_space X]
{f : E → X} (hf : has_compact_mul_support f) (h'f : continuous f) :
f = 1 ∨ finite_dimensional 𝕜 E | begin
by_cases h : ∀ x, f x = 1, { apply or.inl, ext x, exact h x },
apply or.inr,
push_neg at h,
obtain ⟨x, hx⟩ : ∃ x, f x ≠ 1, from h,
have : function.mul_support f ∈ 𝓝 x, from h'f.is_open_mul_support.mem_nhds hx,
obtain ⟨r, rpos, hr⟩ : ∃ (r : ℝ) (hi : 0 < r), metric.closed_ball x r ⊆ function.mul_suppor... | lemma | has_compact_mul_support.eq_one_or_finite_dimensional | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous",
"finite_dimensional",
"finite_dimensional_of_is_compact_closed_ball",
"function.mul_support",
"has_compact_mul_support",
"is_compact",
"is_compact_of_is_closed_subset",
"metric.closed_ball",
"metric.is_closed_ball",
"subset_mul_tsupport",
"t2_space",
"topological_space"
] | If a function has compact multiplicative support, then either the function is trivial or the
space if finite-dimensional. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv.closed_embedding_of_injective {f : E →ₗ[𝕜] F} (hf : f.ker = ⊥)
[finite_dimensional 𝕜 E] :
closed_embedding ⇑f | let g := linear_equiv.of_injective f (linear_map.ker_eq_bot.mp hf) in
{ closed_range := begin
haveI := f.finite_dimensional_range,
simpa [f.range_coe] using f.range.closed_of_finite_dimensional
end,
.. embedding_subtype_coe.comp g.to_continuous_linear_equiv.to_homeomorph.embedding } | lemma | linear_equiv.closed_embedding_of_injective | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"closed_embedding",
"finite_dimensional",
"linear_equiv.of_injective"
] | An injective linear map with finite-dimensional domain is a closed embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.exists_right_inverse_of_surjective [finite_dimensional 𝕜 F]
(f : E →L[𝕜] F) (hf : linear_map.range f = ⊤) :
∃ g : F →L[𝕜] E, f.comp g = continuous_linear_map.id 𝕜 F | let ⟨g, hg⟩ := (f : E →ₗ[𝕜] F).exists_right_inverse_of_surjective hf in
⟨g.to_continuous_linear_map, continuous_linear_map.ext $ linear_map.ext_iff.1 hg⟩ | lemma | continuous_linear_map.exists_right_inverse_of_surjective | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous_linear_map.ext",
"continuous_linear_map.id",
"finite_dimensional",
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_smul_left {c : E} (hc : c ≠ 0) : closed_embedding (λ x : 𝕜, x • c) | linear_equiv.closed_embedding_of_injective (linear_map.ker_to_span_singleton 𝕜 E hc) | lemma | closed_embedding_smul_left | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"closed_embedding",
"linear_equiv.closed_embedding_of_injective",
"linear_map.ker_to_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_smul_left (c : E) : is_closed_map (λ x : 𝕜, x • c) | begin
by_cases hc : c = 0,
{ simp_rw [hc, smul_zero], exact is_closed_map_const },
{ exact (closed_embedding_smul_left hc).is_closed_map }
end | lemma | is_closed_map_smul_left | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"closed_embedding_smul_left",
"is_closed_map",
"is_closed_map_const",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.pi_ring (ι : Type*) [fintype ι] [decidable_eq ι] :
((ι → 𝕜) →L[𝕜] E) ≃L[𝕜] (ι → E) | { continuous_to_fun :=
begin
refine continuous_pi (λ i, _),
exact (continuous_linear_map.apply 𝕜 E (pi.single i 1)).continuous,
end,
continuous_inv_fun :=
begin
simp_rw [linear_equiv.inv_fun_eq_symm, linear_equiv.trans_symm, linear_equiv.symm_symm],
change continuous (linear_map.to_continuous_l... | def | continuous_linear_equiv.pi_ring | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous",
"continuous_linear_map.apply",
"continuous_pi",
"fintype",
"fintype.card",
"linear_equiv.coe_to_linear_map",
"linear_equiv.inv_fun_eq_symm",
"linear_equiv.pi_ring",
"linear_equiv.pi_ring_symm_apply",
"linear_equiv.symm_symm",
"linear_equiv.trans_symm",
"linear_map.coe_comp",
"l... | Continuous linear equivalence between continuous linear functions `𝕜ⁿ → E` and `Eⁿ`.
The spaces `𝕜ⁿ` and `Eⁿ` are represented as `ι → 𝕜` and `ι → E`, respectively,
where `ι` is a finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_clm_apply {X : Type*} [topological_space X]
[finite_dimensional 𝕜 E] {f : X → E →L[𝕜] F} {s : set X} :
continuous_on f s ↔ ∀ y, continuous_on (λ x, f x y) s | begin
refine ⟨λ h y, (continuous_linear_map.apply 𝕜 F y).continuous.comp_continuous_on h, λ h, _⟩,
let d := finrank 𝕜 E,
have hd : d = finrank 𝕜 (fin d → 𝕜) := (finrank_fin_fun 𝕜).symm,
let e₁ : E ≃L[𝕜] fin d → 𝕜 := continuous_linear_equiv.of_finrank_eq hd,
let e₂ : (E →L[𝕜] F) ≃L[𝕜] fin d → F :=
... | lemma | continuous_on_clm_apply | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous.comp_continuous_on",
"continuous_linear_equiv.of_finrank_eq",
"continuous_linear_equiv.pi_ring",
"continuous_linear_map.apply",
"continuous_on",
"finite_dimensional",
"topological_space"
] | A family of continuous linear maps is continuous on `s` if all its applications are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_clm_apply {X : Type*} [topological_space X] [finite_dimensional 𝕜 E]
{f : X → E →L[𝕜] F} :
continuous f ↔ ∀ y, continuous (λ x, f x y) | by simp_rw [continuous_iff_continuous_on_univ, continuous_on_clm_apply] | lemma | continuous_clm_apply | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_clm_apply",
"finite_dimensional",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_dimensional.proper [finite_dimensional 𝕜 E] : proper_space E | begin
set e := continuous_linear_equiv.of_finrank_eq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm,
exact e.symm.antilipschitz.proper_space e.symm.continuous e.symm.surjective
end | lemma | finite_dimensional.proper | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"continuous_linear_equiv.of_finrank_eq",
"finite_dimensional",
"proper_space"
] | Any finite-dimensional vector space over a proper field is proper.
We do not register this as an instance to avoid an instance loop when trying to prove the
properness of `𝕜`, and the search for `𝕜` as an unknown metavariable. Declare the instance
explicitly when needed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_dimensional.proper_real (E : Type u) [normed_add_comm_group E] [normed_space ℝ E]
[finite_dimensional ℝ E] : proper_space E | finite_dimensional.proper ℝ E | instance | finite_dimensional.proper_real | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"finite_dimensional.proper",
"normed_add_comm_group",
"normed_space",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_frontier_inf_dist_compl_eq_dist {E : Type*} [normed_add_comm_group E]
[normed_space ℝ E] [finite_dimensional ℝ E] {x : E} {s : set E} (hx : x ∈ s) (hs : s ≠ univ) :
∃ y ∈ frontier s, metric.inf_dist x sᶜ = dist x y | begin
rcases metric.exists_mem_closure_inf_dist_eq_dist (nonempty_compl.2 hs) x with ⟨y, hys, hyd⟩,
rw closure_compl at hys,
refine ⟨y, ⟨metric.closed_ball_inf_dist_compl_subset_closure hx $
metric.mem_closed_ball.2 $ ge_of_eq _, hys⟩, hyd⟩,
rwa dist_comm
end | lemma | exists_mem_frontier_inf_dist_compl_eq_dist | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"closure_compl",
"dist_comm",
"finite_dimensional",
"frontier",
"ge_of_eq",
"metric.exists_mem_closure_inf_dist_eq_dist",
"metric.inf_dist",
"normed_add_comm_group",
"normed_space"
] | If `E` is a finite dimensional normed real vector space, `x : E`, and `s` is a neighborhood of
`x` that is not equal to the whole space, then there exists a point `y ∈ frontier s` at distance
`metric.inf_dist x sᶜ` from `x`. See also
`is_compact.exists_mem_frontier_inf_dist_compl_eq_dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.exists_mem_frontier_inf_dist_compl_eq_dist {E : Type*} [normed_add_comm_group E]
[normed_space ℝ E] [nontrivial E] {x : E} {K : set E} (hK : is_compact K) (hx : x ∈ K) :
∃ y ∈ frontier K, metric.inf_dist x Kᶜ = dist x y | begin
obtain (hx'|hx') : x ∈ interior K ∪ frontier K,
{ rw ← closure_eq_interior_union_frontier, exact subset_closure hx },
{ rw [mem_interior_iff_mem_nhds, metric.nhds_basis_closed_ball.mem_iff] at hx',
rcases hx' with ⟨r, hr₀, hrK⟩,
haveI : finite_dimensional ℝ E,
from finite_dimensional_of_is_com... | lemma | is_compact.exists_mem_frontier_inf_dist_compl_eq_dist | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"closure_eq_interior_union_frontier",
"dist_self",
"exists_mem_frontier_inf_dist_compl_eq_dist",
"finite_dimensional",
"finite_dimensional_of_is_compact_closed_ball",
"frontier",
"frontier_eq_closure_inter_closure",
"interior",
"is_compact",
"is_compact_of_is_closed_subset",
"mem_interior_iff_me... | If `K` is a compact set in a nontrivial real normed space and `x ∈ K`, then there exists a point
`y` of the boundary of `K` at distance `metric.inf_dist x Kᶜ` from `x`. See also
`exists_mem_frontier_inf_dist_compl_eq_dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_norm_iff {α E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[finite_dimensional ℝ E] {f : α → E} : summable (λ x, ‖f x‖) ↔ summable f | begin
refine ⟨summable_of_summable_norm, λ hf, _⟩,
-- First we use a finite basis to reduce the problem to the case `E = fin N → ℝ`
suffices : ∀ {N : ℕ} {g : α → fin N → ℝ}, summable g → summable (λ x, ‖g x‖),
{ obtain v := fin_basis ℝ E,
set e := v.equiv_funL,
have : summable (λ x, ‖e (f x)‖) := this (... | lemma | summable_norm_iff | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"finset.mem_univ",
"finset.univ",
"norm_norm",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_norm_bounded",
"summable_sum"
] | In a finite dimensional vector space over `ℝ`, the series `∑ x, ‖f x‖` is unconditionally
summable if and only if the series `∑ x, f x` is unconditionally summable. One implication holds in
any complete normed space, while the other holds only in finite dimensional spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_of_is_O' {ι E F : Type*} [normed_add_comm_group E] [complete_space E]
[normed_add_comm_group F] [normed_space ℝ F] [finite_dimensional ℝ F] {f : ι → E} {g : ι → F}
(hg : summable g) (h : f =O[cofinite] g) : summable f | summable_of_is_O (summable_norm_iff.mpr hg) h.norm_right | lemma | summable_of_is_O' | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"complete_space",
"finite_dimensional",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_is_O"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_is_O_nat' {E F : Type*} [normed_add_comm_group E] [complete_space E]
[normed_add_comm_group F] [normed_space ℝ F] [finite_dimensional ℝ F] {f : ℕ → E} {g : ℕ → F}
(hg : summable g) (h : f =O[at_top] g) : summable f | summable_of_is_O_nat (summable_norm_iff.mpr hg) h.norm_right | lemma | summable_of_is_O_nat' | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"complete_space",
"finite_dimensional",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_is_O_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_is_equivalent {ι E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[finite_dimensional ℝ E] {f : ι → E} {g : ι → E}
(hg : summable g) (h : f ~[cofinite] g) : summable f | hg.trans_sub (summable_of_is_O' hg h.is_o.is_O) | lemma | summable_of_is_equivalent | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_is_O'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_is_equivalent_nat {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[finite_dimensional ℝ E] {f : ℕ → E} {g : ℕ → E}
(hg : summable g) (h : f ~[at_top] g) : summable f | hg.trans_sub (summable_of_is_O_nat' hg h.is_o.is_O) | lemma | summable_of_is_equivalent_nat | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_is_O_nat'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.summable_iff {ι E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[finite_dimensional ℝ E] {f : ι → E} {g : ι → E}
(h : f ~[cofinite] g) : summable f ↔ summable g | ⟨λ hf, summable_of_is_equivalent hf h.symm, λ hg, summable_of_is_equivalent hg h⟩ | lemma | is_equivalent.summable_iff | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_is_equivalent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.summable_iff_nat {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[finite_dimensional ℝ E] {f : ℕ → E} {g : ℕ → E}
(h : f ~[at_top] g) : summable f ↔ summable g | ⟨λ hf, summable_of_is_equivalent_nat hf h.symm, λ hg, summable_of_is_equivalent_nat hg h⟩ | lemma | is_equivalent.summable_iff_nat | analysis.normed_space | src/analysis/normed_space/finite_dimension.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.normed_space.add_torsor",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm",
"analysis.normed_space.riesz_lemma",
"topology.algebra.module.finite_dimension",
"topology.algebra.infinite_sum.module",
"topology.instances... | [
"finite_dimensional",
"normed_add_comm_group",
"normed_space",
"summable",
"summable_of_is_equivalent_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_indicator_eq_indicator_norm :
‖indicator s f a‖ = indicator s (λa, ‖f a‖) a | flip congr_fun a (indicator_comp_of_zero norm_zero).symm | lemma | norm_indicator_eq_indicator_norm | analysis.normed_space | src/analysis/normed_space/indicator_function.lean | [
"analysis.normed.group.basic",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_indicator_eq_indicator_nnnorm :
‖indicator s f a‖₊ = indicator s (λa, ‖f a‖₊) a | flip congr_fun a (indicator_comp_of_zero nnnorm_zero).symm | lemma | nnnorm_indicator_eq_indicator_nnnorm | analysis.normed_space | src/analysis/normed_space/indicator_function.lean | [
"analysis.normed.group.basic",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_indicator_le_of_subset (h : s ⊆ t) (f : α → E) (a : α) :
‖indicator s f a‖ ≤ ‖indicator t f a‖ | begin
simp only [norm_indicator_eq_indicator_norm],
exact indicator_le_indicator_of_subset ‹_› (λ _, norm_nonneg _) _
end | lemma | norm_indicator_le_of_subset | analysis.normed_space | src/analysis/normed_space/indicator_function.lean | [
"analysis.normed.group.basic",
"algebra.indicator_function"
] | [
"norm_indicator_eq_indicator_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_norm_le_norm_self : indicator s (λa, ‖f a‖) a ≤ ‖f a‖ | indicator_le_self' (λ _ _, norm_nonneg _) a | lemma | indicator_norm_le_norm_self | analysis.normed_space | src/analysis/normed_space/indicator_function.lean | [
"analysis.normed.group.basic",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_indicator_le_norm_self : ‖indicator s f a‖ ≤ ‖f a‖ | by { rw norm_indicator_eq_indicator_norm, apply indicator_norm_le_norm_self } | lemma | norm_indicator_le_norm_self | analysis.normed_space | src/analysis/normed_space/indicator_function.lean | [
"analysis.normed.group.basic",
"algebra.indicator_function"
] | [
"indicator_norm_le_norm_self",
"norm_indicator_eq_indicator_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 | begin
obtain (rfl|rfl) := int.units_eq_one_or e;
simp only [units.coe_neg_one, units.coe_one, nnnorm_neg, nnnorm_one],
end | lemma | int.nnnorm_coe_units | analysis.normed_space | src/analysis/normed_space/int.lean | [
"analysis.normed.field.basic"
] | [
"int.units_eq_one_or",
"nnnorm_one",
"units.coe_neg_one",
"units.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖ = 1 | by rw [← coe_nnnorm, int.nnnorm_coe_units, nnreal.coe_one] | lemma | int.norm_coe_units | analysis.normed_space | src/analysis/normed_space/int.lean | [
"analysis.normed.field.basic"
] | [
"int.nnnorm_coe_units",
"nnreal.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_coe_nat (n : ℕ) : ‖(n : ℤ)‖₊ = n | real.nnnorm_coe_nat _ | lemma | int.nnnorm_coe_nat | analysis.normed_space | src/analysis/normed_space/int.lean | [
"analysis.normed.field.basic"
] | [
"real.nnnorm_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_add_to_nat_neg_eq_nnnorm (n : ℤ) : ↑(n.to_nat) + ↑((-n).to_nat) = ‖n‖₊ | by rw [← nat.cast_add, to_nat_add_to_nat_neg_eq_nat_abs, nnreal.coe_nat_abs] | lemma | int.to_nat_add_to_nat_neg_eq_nnnorm | analysis.normed_space | src/analysis/normed_space/int.lean | [
"analysis.normed.field.basic"
] | [
"nat.cast_add",
"nnreal.coe_nat_abs",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_add_to_nat_neg_eq_norm (n : ℤ) : ↑(n.to_nat) + ↑((-n).to_nat) = ‖n‖ | by simpa only [nnreal.coe_nat_cast, nnreal.coe_add]
using congr_arg (coe : _ → ℝ) (to_nat_add_to_nat_neg_eq_nnnorm n) | lemma | int.to_nat_add_to_nat_neg_eq_norm | analysis.normed_space | src/analysis/normed_space/int.lean | [
"analysis.normed.field.basic"
] | [
"nnreal.coe_add",
"nnreal.coe_nat_cast",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_R_or_C.norm_coe_norm {z : E} : ‖(‖z‖ : 𝕜)‖ = ‖z‖ | by simp | lemma | is_R_or_C.norm_coe_norm | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul_inv_norm {x : E} (hx : x ≠ 0) : ‖(‖x‖⁻¹ : 𝕜) • x‖ = 1 | begin
have : ‖x‖ ≠ 0 := by simp [hx],
field_simp [norm_smul]
end | lemma | norm_smul_inv_norm | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [
"norm_smul"
] | Lemma to normalize a vector in a normed space `E` over either `ℂ` or `ℝ` to unit length. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_smul_inv_norm' {r : ℝ} (r_nonneg : 0 ≤ r) {x : E} (hx : x ≠ 0) :
‖(r * ‖x‖⁻¹ : 𝕜) • x‖ = r | begin
have : ‖x‖ ≠ 0 := by simp [hx],
field_simp [norm_smul, r_nonneg] with is_R_or_C_simps
end | lemma | norm_smul_inv_norm' | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [
"norm_smul"
] | Lemma to normalize a vector in a normed space `E` over either `ℂ` or `ℝ` to length `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.bound_of_sphere_bound
{r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜) (h : ∀ z ∈ sphere (0 : E) r, ‖f z‖ ≤ c) (z : E) :
‖f z‖ ≤ c / r * ‖z‖ | begin
by_cases z_zero : z = 0,
{ rw z_zero, simp only [linear_map.map_zero, norm_zero, mul_zero], },
set z₁ := (r * ‖z‖⁻¹ : 𝕜) • z with hz₁,
have norm_f_z₁ : ‖f z₁‖ ≤ c,
{ apply h,
rw mem_sphere_zero_iff_norm,
exact norm_smul_inv_norm' r_pos.le z_zero },
have r_ne_zero : (r : 𝕜) ≠ 0 := is_R_or_C.o... | lemma | linear_map.bound_of_sphere_bound | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [
"div_le_div",
"div_mul_cancel",
"div_mul_eq_mul_div",
"is_R_or_C.norm_coe_norm",
"is_R_or_C.norm_of_nonneg",
"is_R_or_C.of_real_eq_zero",
"linear_map.map_smul",
"linear_map.map_zero",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_le_mul",
"mul_zero",
"norm_div",
"norm_eq_zero",
"norm... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.bound_of_ball_bound' {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜)
(h : ∀ z ∈ closed_ball (0 : E) r, ‖f z‖ ≤ c) (z : E) :
‖f z‖ ≤ c / r * ‖z‖ | f.bound_of_sphere_bound r_pos c (λ z hz, h z hz.le) z | lemma | linear_map.bound_of_ball_bound' | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [] | `linear_map.bound_of_ball_bound` is a version of this over arbitrary nontrivially normed fields.
It produces a less precise bound so we keep both versions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.op_norm_bound_of_ball_bound
{r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →L[𝕜] 𝕜) (h : ∀ z ∈ closed_ball (0 : E) r, ‖f z‖ ≤ c) :
‖f‖ ≤ c / r | begin
apply continuous_linear_map.op_norm_le_bound,
{ apply div_nonneg _ r_pos.le,
exact (norm_nonneg _).trans
(h 0 (by simp only [norm_zero, mem_closed_ball, dist_zero_left, r_pos.le])), },
apply linear_map.bound_of_ball_bound' r_pos,
exact λ z hz, h z hz,
end | lemma | continuous_linear_map.op_norm_bound_of_ball_bound | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [
"continuous_linear_map.op_norm_le_bound",
"div_nonneg",
"linear_map.bound_of_ball_bound'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space.sphere_nonempty_is_R_or_C [nontrivial E] {r : ℝ} (hr : 0 ≤ r) :
nonempty (sphere (0:E) r) | begin
letI : normed_space ℝ E := normed_space.restrict_scalars ℝ 𝕜 E,
exact (normed_space.sphere_nonempty.mpr hr).coe_sort,
end | lemma | normed_space.sphere_nonempty_is_R_or_C | analysis.normed_space | src/analysis/normed_space/is_R_or_C.lean | [
"data.is_R_or_C.basic",
"analysis.normed_space.operator_norm",
"analysis.normed_space.pointwise"
] | [
"nontrivial",
"normed_space",
"normed_space.restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry (σ₁₂ : R →+* R₂) (E E₂ : Type*) [seminormed_add_comm_group E]
[seminormed_add_comm_group E₂] [module R E] [module R₂ E₂] extends E →ₛₗ[σ₁₂] E₂ | (norm_map' : ∀ x, ‖to_linear_map x‖ = ‖x‖) | structure | linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"seminormed_add_comm_group"
] | A `σ₁₂`-semilinear isometric embedding of a normed `R`-module into an `R₂`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semilinear_isometry_class (𝓕 : Type*) {R R₂ : out_param Type*} [semiring R] [semiring R₂]
(σ₁₂ : out_param $ R →+* R₂) (E E₂ : out_param Type*) [seminormed_add_comm_group E]
[seminormed_add_comm_group E₂] [module R E] [module R₂ E₂]
extends semilinear_map_class 𝓕 σ₁₂ E E₂ | (norm_map : ∀ (f : 𝓕) (x : E), ‖f x‖ = ‖x‖) | class | semilinear_isometry_class | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"semilinear_map_class",
"seminormed_add_comm_group",
"semiring"
] | `semilinear_isometry_class F σ E E₂` asserts `F` is a type of bundled `σ`-semilinear isometries
`E → E₂`.
See also `linear_isometry_class F R E E₂` for the case where `σ` is the identity map on `R`.
A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S`
is semilinear if it satisfies ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_class (𝓕 : Type*) (R E E₂ : out_param Type*) [semiring R]
[seminormed_add_comm_group E] [seminormed_add_comm_group E₂] [module R E] [module R E₂] | semilinear_isometry_class 𝓕 (ring_hom.id R) E E₂ | abbreviation | linear_isometry_class | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"ring_hom.id",
"semilinear_isometry_class",
"seminormed_add_comm_group",
"semiring"
] | `linear_isometry_class F R E E₂` asserts `F` is a type of bundled `R`-linear isometries
`M → M₂`.
This is an abbreviation for `semilinear_isometry_class F (ring_hom.id R) E E₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : isometry f | add_monoid_hom_class.isometry_of_norm _ (norm_map _) | lemma | semilinear_isometry_class.isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry",
"semilinear_isometry_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) :
continuous f | (semilinear_isometry_class.isometry f).continuous | lemma | semilinear_isometry_class.continuous | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"continuous",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_map [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E) :
‖f x‖₊ = ‖x‖₊ | nnreal.eq $ norm_map f x | lemma | semilinear_isometry_class.nnnorm_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"nnreal.eq",
"semilinear_isometry_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) :
lipschitz_with 1 f | (semilinear_isometry_class.isometry f).lipschitz | lemma | semilinear_isometry_class.lipschitz | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"lipschitz_with",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) :
antilipschitz_with 1 f | (semilinear_isometry_class.isometry f).antilipschitz | lemma | semilinear_isometry_class.antilipschitz | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"antilipschitz_with",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : set E) :
emetric.diam (f '' s) = emetric.diam s | (semilinear_isometry_class.isometry f).ediam_image s | lemma | semilinear_isometry_class.ediam_image | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"emetric.diam",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_range [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) :
emetric.diam (range f) = emetric.diam (univ : set E) | (semilinear_isometry_class.isometry f).ediam_range | lemma | semilinear_isometry_class.ediam_range | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"emetric.diam",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_image [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : set E) :
metric.diam (f '' s) = metric.diam s | (semilinear_isometry_class.isometry f).diam_image s | lemma | semilinear_isometry_class.diam_image | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.diam",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_range [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) :
metric.diam (range f) = metric.diam (univ : set E) | (semilinear_isometry_class.isometry f).diam_range | lemma | semilinear_isometry_class.diam_range | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.diam",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_injective : injective (to_linear_map : (E →ₛₗᵢ[σ₁₂] E₂) → (E →ₛₗ[σ₁₂] E₂)) | | ⟨f, _⟩ ⟨g, _⟩ rfl := rfl | lemma | linear_isometry.to_linear_map_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_inj {f g : E →ₛₗᵢ[σ₁₂] E₂} :
f.to_linear_map = g.to_linear_map ↔ f = g | to_linear_map_injective.eq_iff | lemma | linear_isometry.to_linear_map_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_map : ⇑f.to_linear_map = f | rfl | lemma | linear_isometry.coe_to_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : E →ₛₗ[σ₁₂] E₂) (hf) : ⇑(mk f hf) = f | rfl | lemma | linear_isometry.coe_mk | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @injective (E →ₛₗᵢ[σ₁₂] E₂) (E → E₂) coe_fn | fun_like.coe_injective | lemma | linear_isometry.coe_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (σ₁₂ : R →+* R₂) (E E₂ : Type*) [seminormed_add_comm_group E]
[seminormed_add_comm_group E₂] [module R E] [module R₂ E₂] (h : E →ₛₗᵢ[σ₁₂] E₂) : E → E₂ | h
initialize_simps_projections linear_isometry (to_linear_map_to_fun → apply) | def | linear_isometry.simps.apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_isometry",
"module",
"seminormed_add_comm_group"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : E →ₛₗᵢ[σ₁₂] E₂} (h : ∀ x, f x = g x) : f = g | coe_injective $ funext h | lemma | linear_isometry.ext | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg [semilinear_isometry_class 𝓕 σ₁₂ E E₂] {f : 𝓕} :
Π {x x' : E}, x = x' → f x = f x' | | _ _ rfl := rfl | lemma | linear_isometry.congr_arg | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"semilinear_isometry_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun [semilinear_isometry_class 𝓕 σ₁₂ E E₂] {f g : 𝓕} (h : f = g) (x : E) :
f x = g x | h ▸ rfl | lemma | linear_isometry.congr_fun | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"semilinear_isometry_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : f 0 = 0 | f.to_linear_map.map_zero | lemma | linear_isometry.map_zero | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (x y : E) : f (x + y) = f x + f y | f.to_linear_map.map_add x y | lemma | linear_isometry.map_add | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x : E) : f (- x) = - f x | f.to_linear_map.map_neg x | lemma | linear_isometry.map_neg | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y : E) : f (x - y) = f x - f y | f.to_linear_map.map_sub x y | lemma | linear_isometry.map_sub | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ (c : R) (x : E) : f (c • x) = σ₁₂ c • f x | f.to_linear_map.map_smulₛₗ c x | lemma | linear_isometry.map_smulₛₗ | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul [module R E₂] (f : E →ₗᵢ[R] E₂) (c : R) (x : E) :
f (c • x) = c • f x | f.to_linear_map.map_smul c x | lemma | linear_isometry.map_smul | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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