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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coe_injective : @function.injective (normed_add_group_hom V₁ V₂) (V₁ → V₂) coe_fn | by apply coe_inj | lemma | normed_add_group_hom.coe_injective | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj_iff : f = g ↔ (f : V₁ → V₂) = g | ⟨congr_arg _, coe_inj⟩ | lemma | normed_add_group_hom.coe_inj_iff | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (H : ∀ x, f x = g x) : f = g | coe_inj $ funext H | lemma | normed_add_group_hom.ext | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff : f = g ↔ ∀ x, f x = g x | ⟨by rintro rfl x; refl, ext⟩ | lemma | normed_add_group_hom.ext_iff | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe : f.to_fun = f | rfl | lemma | normed_add_group_hom.to_fun_eq_coe | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f) (h₁) (h₂) (h₃) : ⇑(⟨f, h₁, h₂, h₃⟩ : normed_add_group_hom V₁ V₂) = f | rfl | lemma | normed_add_group_hom.coe_mk | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk_normed_add_group_hom (f : V₁ →+ V₂) (C) (hC) :
⇑(f.mk_normed_add_group_hom C hC) = f | rfl | lemma | normed_add_group_hom.coe_mk_normed_add_group_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk_normed_add_group_hom' (f : V₁ →+ V₂) (C) (hC) :
⇑(f.mk_normed_add_group_hom' C hC) = f | rfl | lemma | normed_add_group_hom.coe_mk_normed_add_group_hom' | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom (f : normed_add_group_hom V₁ V₂) : V₁ →+ V₂ | add_monoid_hom.mk' f f.map_add' | def | normed_add_group_hom.to_add_monoid_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | The group homomorphism underlying a bounded group homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_add_monoid_hom : ⇑f.to_add_monoid_hom = f | rfl | lemma | normed_add_group_hom.coe_to_add_monoid_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom_injective :
function.injective (@normed_add_group_hom.to_add_monoid_hom V₁ V₂ _ _) | λ f g h, coe_inj $ show ⇑f.to_add_monoid_hom = g, by { rw h, refl } | lemma | normed_add_group_hom.to_add_monoid_hom_injective | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom.to_add_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_add_monoid_hom (f) (h₁) (h₂) :
(⟨f, h₁, h₂⟩ : normed_add_group_hom V₁ V₂).to_add_monoid_hom = add_monoid_hom.mk' f h₁ | rfl | lemma | normed_add_group_hom.mk_to_add_monoid_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bound : ∃ C, 0 < C ∧ ∀ x, ‖f x‖ ≤ C * ‖x‖ | let ⟨C, hC⟩ := f.bound' in exists_pos_bound_of_bound _ hC | lemma | normed_add_group_hom.bound | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound",
"exists_pos_bound_of_bound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_of_norm_ge {K : ℝ≥0} (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
antilipschitz_with K f | antilipschitz_with.of_le_mul_dist $
λ x y, by simpa only [dist_eq_norm, map_sub] using h (x - y) | theorem | normed_add_group_hom.antilipschitz_of_norm_ge | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_on_with (f : normed_add_group_hom V₁ V₂) (K : add_subgroup V₂) (C : ℝ) : Prop | ∀ h ∈ K, ∃ g, f g = h ∧ ‖g‖ ≤ C*‖h‖ | def | normed_add_group_hom.surjective_on_with | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup",
"normed_add_group_hom"
] | A normed group hom is surjective on the subgroup `K` with constant `C` if every element
`x` of `K` has a preimage whose norm is bounded above by `C*‖x‖`. This is a more
abstract version of `f` having a right inverse defined on `K` with operator norm
at most `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_on_with.mono {f : normed_add_group_hom V₁ V₂} {K : add_subgroup V₂} {C C' : ℝ}
(h : f.surjective_on_with K C) (H : C ≤ C') : f.surjective_on_with K C' | begin
intros k k_in,
rcases h k k_in with ⟨g, rfl, hg⟩,
use [g, rfl],
by_cases Hg : ‖f g‖ = 0,
{ simpa [Hg] using hg },
{ exact hg.trans ((mul_le_mul_right $ (ne.symm Hg).le_iff_lt.mp (norm_nonneg _)).mpr H) }
end | lemma | normed_add_group_hom.surjective_on_with.mono | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup",
"mul_le_mul_right",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_on_with.exists_pos {f : normed_add_group_hom V₁ V₂} {K : add_subgroup V₂} {C : ℝ}
(h : f.surjective_on_with K C) : ∃ C' > 0, f.surjective_on_with K C' | begin
refine ⟨|C| + 1, _, _⟩,
{ linarith [abs_nonneg C] },
{ apply h.mono,
linarith [le_abs_self C] }
end | lemma | normed_add_group_hom.surjective_on_with.exists_pos | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"abs_nonneg",
"add_subgroup",
"le_abs_self",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_on_with.surj_on {f : normed_add_group_hom V₁ V₂} {K : add_subgroup V₂} {C : ℝ}
(h : f.surjective_on_with K C) : set.surj_on f set.univ K | λ x hx, (h x hx).imp $ λ a ⟨ha, _⟩, ⟨set.mem_univ _, ha⟩ | lemma | normed_add_group_hom.surjective_on_with.surj_on | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup",
"normed_add_group_hom",
"set.surj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm (f : normed_add_group_hom V₁ V₂) | Inf {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} | def | normed_add_group_hom.op_norm | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | The operator norm of a seminormed group homomorphism is the inf of all its bounds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_op_norm : has_norm (normed_add_group_hom V₁ V₂) | ⟨op_norm⟩ | instance | normed_add_group_hom.has_op_norm | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"has_norm",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_def : ‖f‖ = Inf {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} | rfl | lemma | normed_add_group_hom.norm_def | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_nonempty {f : normed_add_group_hom V₁ V₂} :
∃ c, c ∈ { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } | let ⟨M, hMp, hMb⟩ := f.bound in ⟨M, le_of_lt hMp, hMb⟩ | lemma | normed_add_group_hom.bounds_nonempty | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_bdd_below {f : normed_add_group_hom V₁ V₂} :
bdd_below {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} | ⟨0, λ _ ⟨hn, _⟩, hn⟩ | lemma | normed_add_group_hom.bounds_bdd_below | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bdd_below",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_nonneg : 0 ≤ ‖f‖ | le_cInf bounds_nonempty (λ _ ⟨hx, _⟩, hx) | lemma | normed_add_group_hom.op_norm_nonneg | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"le_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_op_norm (x : V₁) : ‖f x‖ ≤ ‖f‖ * ‖x‖ | begin
obtain ⟨C, Cpos, hC⟩ := f.bound,
replace hC := hC x,
by_cases h : ‖x‖ = 0,
{ rwa [h, mul_zero] at ⊢ hC },
have hlt : 0 < ‖x‖ := lt_of_le_of_ne (norm_nonneg x) (ne.symm h),
exact (div_le_iff hlt).mp (le_cInf bounds_nonempty (λ c ⟨_, hc⟩,
(div_le_iff hlt).mpr $ by { apply hc })),
end | theorem | normed_add_group_hom.le_op_norm | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"div_le_iff",
"le_cInf",
"mul_zero"
] | The fundamental property of the operator norm: `‖f x‖ ≤ ‖f‖ * ‖x‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_op_norm_of_le {c : ℝ} {x} (h : ‖x‖ ≤ c) : ‖f x‖ ≤ ‖f‖ * c | le_trans (f.le_op_norm x) (mul_le_mul_of_nonneg_left h f.op_norm_nonneg) | theorem | normed_add_group_hom.le_op_norm_of_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_op_norm_le {c : ℝ} (h : ‖f‖ ≤ c) (x : V₁) : ‖f x‖ ≤ c * ‖x‖ | (f.le_op_norm x).trans (mul_le_mul_of_nonneg_right h (norm_nonneg x)) | theorem | normed_add_group_hom.le_of_op_norm_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz : lipschitz_with ⟨‖f‖, op_norm_nonneg f⟩ f | lipschitz_with.of_dist_le_mul $ λ x y,
by { rw [dist_eq_norm, dist_eq_norm, ←map_sub], apply le_op_norm } | theorem | normed_add_group_hom.lipschitz | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"lipschitz_with"
] | continuous linear maps are Lipschitz continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous (f : normed_add_group_hom V₁ V₂) :
uniform_continuous f | f.lipschitz.uniform_continuous | lemma | normed_add_group_hom.uniform_continuous | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (f : normed_add_group_hom V₁ V₂) : continuous f | f.uniform_continuous.continuous | lemma | normed_add_group_hom.continuous | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"continuous",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ratio_le_op_norm (x : V₁) : ‖f x‖ / ‖x‖ ≤ ‖f‖ | div_le_of_nonneg_of_le_mul (norm_nonneg _) f.op_norm_nonneg (le_op_norm _ _) | lemma | normed_add_group_hom.ratio_le_op_norm | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"div_le_of_nonneg_of_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_bound {M : ℝ} (hMp: 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) :
‖f‖ ≤ M | cInf_le bounds_bdd_below ⟨hMp, hM⟩ | lemma | normed_add_group_hom.op_norm_le_bound | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"cInf_le"
] | If one controls the norm of every `f x`, then one controls the norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_eq_of_bounds {M : ℝ} (M_nonneg : 0 ≤ M)
(h_above : ∀ x, ‖f x‖ ≤ M*‖x‖) (h_below : ∀ N ≥ 0, (∀ x, ‖f x‖ ≤ N*‖x‖) → M ≤ N) :
‖f‖ = M | le_antisymm (f.op_norm_le_bound M_nonneg h_above)
((le_cInf_iff normed_add_group_hom.bounds_bdd_below ⟨M, M_nonneg, h_above⟩).mpr $
λ N ⟨N_nonneg, hN⟩, h_below N N_nonneg hN) | lemma | normed_add_group_hom.op_norm_eq_of_bounds | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"le_cInf_iff",
"normed_add_group_hom.bounds_bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_of_lipschitz {f : normed_add_group_hom V₁ V₂} {K : ℝ≥0}
(hf : lipschitz_with K f) :
‖f‖ ≤ K | f.op_norm_le_bound K.2 $ λ x, by simpa only [dist_zero_right, map_zero] using hf.dist_le_mul x 0 | theorem | normed_add_group_hom.op_norm_le_of_lipschitz | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"lipschitz_with",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_normed_add_group_hom_norm_le (f : V₁ →+ V₂) {C : ℝ} (hC : 0 ≤ C)
(h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mk_normed_add_group_hom C h‖ ≤ C | op_norm_le_bound _ hC h | lemma | normed_add_group_hom.mk_normed_add_group_hom_norm_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | If a bounded group homomorphism map is constructed from a group homomorphism via the constructor
`mk_normed_add_group_hom`, then its norm is bounded by the bound given to the constructor if it is
nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_normed_add_group_hom_norm_le' (f : V₁ →+ V₂) {C : ℝ} (h : ∀x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mk_normed_add_group_hom C h‖ ≤ max C 0 | op_norm_le_bound _ (le_max_right _ _) $ λ x, (h x).trans $
mul_le_mul_of_nonneg_right (le_max_left _ _) (norm_nonneg x) | lemma | normed_add_group_hom.mk_normed_add_group_hom_norm_le' | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"mul_le_mul_of_nonneg_right"
] | If a bounded group homomorphism map is constructed from a group homomorphism
via the constructor `mk_normed_add_group_hom`, then its norm is bounded by the bound
given to the constructor or zero if this bound is negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ | mk_normed_add_group_hom_norm_le _ (add_nonneg (op_norm_nonneg _) (op_norm_nonneg _)) _ | theorem | normed_add_group_hom.op_norm_add_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | The operator norm satisfies the triangle inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add (f g : normed_add_group_hom V₁ V₂) : ⇑(f + g) = (f + g : V₁ → V₂) | rfl | lemma | normed_add_group_hom.coe_add | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (f g : normed_add_group_hom V₁ V₂) (v : V₁) :
(f + g : normed_add_group_hom V₁ V₂) v = f v + g v | rfl | lemma | normed_add_group_hom.add_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_zero : ‖(0 : normed_add_group_hom V₁ V₂)‖ = 0 | le_antisymm (cInf_le bounds_bdd_below
⟨ge_of_eq rfl, λ _, le_of_eq (by { rw [zero_mul], exact norm_zero })⟩)
(op_norm_nonneg _) | theorem | normed_add_group_hom.op_norm_zero | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"cInf_le",
"normed_add_group_hom",
"zero_mul"
] | The norm of the `0` operator is `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_zero_iff {V₁ V₂ : Type*} [normed_add_comm_group V₁] [normed_add_comm_group V₂]
{f : normed_add_group_hom V₁ V₂} : ‖f‖ = 0 ↔ f = 0 | iff.intro
(λ hn, ext (λ x, norm_le_zero_iff.1
(calc _ ≤ ‖f‖ * ‖x‖ : le_op_norm _ _
... = _ : by rw [hn, zero_mul])))
(λ hf, by rw [hf, op_norm_zero] ) | theorem | normed_add_group_hom.op_norm_zero_iff | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_comm_group",
"normed_add_group_hom",
"zero_mul"
] | For normed groups, an operator is zero iff its norm vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero : ⇑(0 : normed_add_group_hom V₁ V₂) = (0 : V₁ → V₂) | rfl | lemma | normed_add_group_hom.coe_zero | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (v : V₁) : (0 : normed_add_group_hom V₁ V₂) v = 0 | rfl | lemma | normed_add_group_hom.zero_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : normed_add_group_hom V V | (add_monoid_hom.id V).mk_normed_add_group_hom 1 (by simp [le_refl]) | def | normed_add_group_hom.id | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | The identity as a continuous normed group hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_id_le : ‖(id V : normed_add_group_hom V V)‖ ≤ 1 | op_norm_le_bound _ zero_le_one (λx, by simp) | lemma | normed_add_group_hom.norm_id_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom",
"zero_le_one"
] | The norm of the identity is at most `1`. It is in fact `1`, except when the norm of every
element vanishes, where it is `0`. (Since we are working with seminorms this can happen even if the
space is non-trivial.) It means that one can not do better than an inequality in general. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_id_of_nontrivial_seminorm (h : ∃ (x : V), ‖x‖ ≠ 0 ) :
‖(id V)‖ = 1 | le_antisymm (norm_id_le V) $ let ⟨x, hx⟩ := h in
have _ := (id V).ratio_le_op_norm x,
by rwa [id_apply, div_self hx] at this | lemma | normed_add_group_hom.norm_id_of_nontrivial_seminorm | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"div_self"
] | If there is an element with norm different from `0`, then the norm of the identity equals `1`.
(Since we are working with seminorms supposing that the space is non-trivial is not enough.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_id {V : Type*} [normed_add_comm_group V] [nontrivial V] : ‖(id V)‖ = 1 | begin
refine norm_id_of_nontrivial_seminorm V _,
obtain ⟨x, hx⟩ := exists_ne (0 : V),
exact ⟨x, ne_of_gt (norm_pos_iff.2 hx)⟩,
end | lemma | normed_add_group_hom.norm_id | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"exists_ne",
"nontrivial",
"normed_add_comm_group"
] | If a normed space is non-trivial, then the norm of the identity equals `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ((normed_add_group_hom.id V) : V → V) = (_root_.id : V → V) | rfl | lemma | normed_add_group_hom.coe_id | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (f : normed_add_group_hom V₁ V₂) : ⇑(-f) = (-f : V₁ → V₂) | rfl | lemma | normed_add_group_hom.coe_neg | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply (f : normed_add_group_hom V₁ V₂) (v : V₁) :
(-f : normed_add_group_hom V₁ V₂) v = - (f v) | rfl | lemma | normed_add_group_hom.neg_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_neg (f : normed_add_group_hom V₁ V₂) : ‖-f‖ = ‖f‖ | by simp only [norm_def, coe_neg, norm_neg, pi.neg_apply] | lemma | normed_add_group_hom.op_norm_neg | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (f g : normed_add_group_hom V₁ V₂) : ⇑(f - g) = (f - g : V₁ → V₂) | rfl | lemma | normed_add_group_hom.coe_sub | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (f g : normed_add_group_hom V₁ V₂) (v : V₁) :
(f - g : normed_add_group_hom V₁ V₂) v = f v - g v | rfl | lemma | normed_add_group_hom.sub_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (r : R) (f : normed_add_group_hom V₁ V₂) : ⇑(r • f) = r • f | rfl | lemma | normed_add_group_hom.coe_smul | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (r : R) (f : normed_add_group_hom V₁ V₂) (v : V₁) : (r • f) v = r • f v | rfl | lemma | normed_add_group_hom.smul_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_scalar : has_smul ℕ (normed_add_group_hom V₁ V₂) | { smul := λ n f,
{ to_fun := n • f,
map_add' := (n • f.to_add_monoid_hom).map_add',
bound' := let ⟨b, hb⟩ := f.bound' in ⟨n • b, λ v, begin
rw [pi.smul_apply, nsmul_eq_mul, mul_assoc],
exact (norm_nsmul_le _ _).trans (mul_le_mul_of_nonneg_left (hb _) (nat.cast_nonneg _)),
end⟩ } } | instance | normed_add_group_hom.has_nat_scalar | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"has_smul",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"nat.cast_nonneg",
"normed_add_group_hom",
"nsmul_eq_mul",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (r : ℕ) (f : normed_add_group_hom V₁ V₂) : ⇑(r • f) = r • f | rfl | lemma | normed_add_group_hom.coe_nsmul | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_apply (r : ℕ) (f : normed_add_group_hom V₁ V₂) (v : V₁) : (r • f) v = r • f v | rfl | lemma | normed_add_group_hom.nsmul_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_int_scalar : has_smul ℤ (normed_add_group_hom V₁ V₂) | { smul := λ z f,
{ to_fun := z • f,
map_add' := (z • f.to_add_monoid_hom).map_add',
bound' := let ⟨b, hb⟩ := f.bound' in ⟨‖z‖ • b, λ v, begin
rw [pi.smul_apply, smul_eq_mul, mul_assoc],
exact (norm_zsmul_le _ _).trans (mul_le_mul_of_nonneg_left (hb _) $ norm_nonneg _),
end⟩ } } | instance | normed_add_group_hom.has_int_scalar | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"has_smul",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"normed_add_group_hom",
"pi.smul_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zsmul (r : ℤ) (f : normed_add_group_hom V₁ V₂) : ⇑(r • f) = r • f | rfl | lemma | normed_add_group_hom.coe_zsmul | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_apply (r : ℤ) (f : normed_add_group_hom V₁ V₂) (v : V₁) : (r • f) v = r • f v | rfl | lemma | normed_add_group_hom.zsmul_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminormed_add_comm_group : seminormed_add_comm_group (normed_add_group_hom V₁ V₂) | add_group_seminorm.to_seminormed_add_comm_group
{ to_fun := op_norm,
map_zero' := op_norm_zero,
neg' := op_norm_neg,
add_le' := op_norm_add_le } | instance | normed_add_group_hom.to_seminormed_add_comm_group | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom",
"seminormed_add_comm_group"
] | Normed group homomorphisms themselves form a seminormed group with respect to
the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_add_comm_group {V₁ V₂ : Type*} [normed_add_comm_group V₁]
[normed_add_comm_group V₂] :
normed_add_comm_group (normed_add_group_hom V₁ V₂) | add_group_norm.to_normed_add_comm_group
{ to_fun := op_norm,
map_zero' := op_norm_zero,
neg' := op_norm_neg,
add_le' := op_norm_add_le,
eq_zero_of_map_eq_zero' := λ f, op_norm_zero_iff.1 } | instance | normed_add_group_hom.to_normed_add_comm_group | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_comm_group",
"normed_add_group_hom"
] | Normed group homomorphisms themselves form a normed group with respect to
the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_add_hom : normed_add_group_hom V₁ V₂ →+ (V₁ → V₂) | { to_fun := coe_fn, map_zero' := coe_zero, map_add' := coe_add} | def | normed_add_group_hom.coe_fn_add_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | Coercion of a `normed_add_group_hom` is an `add_monoid_hom`. Similar to `add_monoid_hom.coe_fn`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sum {ι : Type*} (s : finset ι) (f : ι → normed_add_group_hom V₁ V₂) :
⇑(∑ i in s, f i) = ∑ i in s, (f i) | (coe_fn_add_hom : _ →+ (V₁ → V₂)).map_sum f s | lemma | normed_add_group_hom.coe_sum | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"finset",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_apply {ι : Type*} (s : finset ι) (f : ι → normed_add_group_hom V₁ V₂) (v : V₁) :
(∑ i in s, f i) v = ∑ i in s, (f i v) | by simp only [coe_sum, finset.sum_apply] | lemma | normed_add_group_hom.sum_apply | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"finset",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : normed_add_group_hom V₂ V₃) (f : normed_add_group_hom V₁ V₂) :
normed_add_group_hom V₁ V₃ | (g.to_add_monoid_hom.comp f.to_add_monoid_hom).mk_normed_add_group_hom (‖g‖ * ‖f‖) $ λ v, calc
‖g (f v)‖ ≤ ‖g‖ * ‖f v‖ : le_op_norm _ _
... ≤ ‖g‖ * (‖f‖ * ‖v‖) : mul_le_mul_of_nonneg_left (le_op_norm _ _) (op_norm_nonneg _)
... = ‖g‖ * ‖f‖ * ‖v‖ : by rw mul_assoc | def | normed_add_group_hom.comp | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"normed_add_group_hom"
] | The composition of continuous normed group homs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_comp_le (g : normed_add_group_hom V₂ V₃) (f : normed_add_group_hom V₁ V₂) :
‖g.comp f‖ ≤ ‖g‖ * ‖f‖ | mk_normed_add_group_hom_norm_le _ (mul_nonneg (op_norm_nonneg _) (op_norm_nonneg _)) _ | lemma | normed_add_group_hom.norm_comp_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_le_of_le {g : normed_add_group_hom V₂ V₃} {C₁ C₂ : ℝ} (hg : ‖g‖ ≤ C₂)
(hf : ‖f‖ ≤ C₁) :
‖g.comp f‖ ≤ C₂ * C₁ | le_trans (norm_comp_le g f) $ mul_le_mul hg hf (norm_nonneg _) (le_trans (norm_nonneg _) hg) | lemma | normed_add_group_hom.norm_comp_le_of_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"mul_le_mul",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_le_of_le' {g : normed_add_group_hom V₂ V₃} (C₁ C₂ C₃ : ℝ) (h : C₃ = C₂ * C₁)
(hg : ‖g‖ ≤ C₂) (hf : ‖f‖ ≤ C₁) : ‖g.comp f‖ ≤ C₃ | by { rw h, exact norm_comp_le_of_le hg hf } | lemma | normed_add_group_hom.norm_comp_le_of_le' | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom :
normed_add_group_hom V₂ V₃ →+ normed_add_group_hom V₁ V₂ →+ normed_add_group_hom V₁ V₃ | add_monoid_hom.mk' (λ g, add_monoid_hom.mk' (λ f, g.comp f)
(by { intros, ext, exact map_add g _ _ }))
(by { intros, ext, simp only [comp_apply, pi.add_apply, function.comp_app,
add_monoid_hom.add_apply, add_monoid_hom.mk'_apply, coe_add] }) | def | normed_add_group_hom.comp_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | Composition of normed groups hom as an additive group morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_zero (f : normed_add_group_hom V₂ V₃) :
f.comp (0 : normed_add_group_hom V₁ V₂) = 0 | by { ext, exact map_zero f } | lemma | normed_add_group_hom.comp_zero | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp (f : normed_add_group_hom V₁ V₂) :
(0 : normed_add_group_hom V₂ V₃).comp f = 0 | by { ext, refl } | lemma | normed_add_group_hom.zero_comp | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc {V₄: Type* } [seminormed_add_comm_group V₄] (h : normed_add_group_hom V₃ V₄)
(g : normed_add_group_hom V₂ V₃) (f : normed_add_group_hom V₁ V₂) :
(h.comp g).comp f = h.comp (g.comp f) | by { ext, refl } | lemma | normed_add_group_hom.comp_assoc | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp (f : normed_add_group_hom V₁ V₂) (g : normed_add_group_hom V₂ V₃) :
(g.comp f : V₁ → V₃) = (g : V₂ → V₃) ∘ (f : V₁ → V₂) | rfl | lemma | normed_add_group_hom.coe_comp | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl (s : add_subgroup V) : normed_add_group_hom s V | { to_fun := (coe : s → V),
map_add' := λ v w, add_subgroup.coe_add _ _ _,
bound' := ⟨1, λ v, by { rw [one_mul], refl }⟩ } | def | normed_add_group_hom.incl | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup",
"bound'",
"normed_add_group_hom",
"one_mul"
] | The inclusion of an `add_subgroup`, as bounded group homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_incl {V' : add_subgroup V} (x : V') : ‖incl _ x‖ = ‖x‖ | rfl | lemma | normed_add_group_hom.norm_incl | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker : add_subgroup V₁ | f.to_add_monoid_hom.ker | def | normed_add_group_hom.ker | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup"
] | The kernel of a bounded group homomorphism. Naturally endowed with a
`seminormed_add_comm_group` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ker (v : V₁) : v ∈ f.ker ↔ f v = 0 | by { erw f.to_add_monoid_hom.mem_ker, refl } | lemma | normed_add_group_hom.mem_ker | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker.lift (h : g.comp f = 0) :
normed_add_group_hom V₁ g.ker | { to_fun := λ v, ⟨f v, by { erw g.mem_ker, show (g.comp f) v = 0, rw h, refl }⟩,
map_add' := λ v w, by { simp only [map_add], refl },
bound' := f.bound' } | def | normed_add_group_hom.ker.lift | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"normed_add_group_hom"
] | Given a normed group hom `f : V₁ → V₂` satisfying `g.comp f = 0` for some `g : V₂ → V₃`,
the corestriction of `f` to the kernel of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker.incl_comp_lift (h : g.comp f = 0) :
(incl g.ker).comp (ker.lift f g h) = f | by { ext, refl } | lemma | normed_add_group_hom.ker.incl_comp_lift | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_zero : (0 : normed_add_group_hom V₁ V₂).ker = ⊤ | by { ext, simp [mem_ker] } | lemma | normed_add_group_hom.ker_zero | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ker : (f.ker : set V₁) = (f : V₁ → V₂) ⁻¹' {0} | rfl | lemma | normed_add_group_hom.coe_ker | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_ker {V₂ : Type*} [normed_add_comm_group V₂] (f : normed_add_group_hom V₁ V₂) :
is_closed (f.ker : set V₁) | f.coe_ker ▸ is_closed.preimage f.continuous (t1_space.t1 0) | lemma | normed_add_group_hom.is_closed_ker | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"is_closed",
"is_closed.preimage",
"normed_add_comm_group",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range : add_subgroup V₂ | f.to_add_monoid_hom.range | def | normed_add_group_hom.range | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup"
] | The image of a bounded group homomorphism. Naturally endowed with a
`seminormed_add_comm_group` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_range (v : V₂) : v ∈ f.range ↔ ∃ w, f w = v | by { rw [range, add_monoid_hom.mem_range], refl } | lemma | normed_add_group_hom.mem_range | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_self (v : V₁) : f v ∈ f.range | ⟨v, rfl⟩ | lemma | normed_add_group_hom.mem_range_self | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_range : (g.comp f).range = add_subgroup.map g.to_add_monoid_hom f.range | by { erw add_monoid_hom.map_range, refl } | lemma | normed_add_group_hom.comp_range | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_range (s : add_subgroup V₁) : (incl s).range = s | by { ext x, exact ⟨λ ⟨y, hy⟩, by { rw ← hy; simp }, λ hx, ⟨⟨x, hx⟩, by simp⟩⟩ } | lemma | normed_add_group_hom.incl_range | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_comp_incl_top : (f.comp (incl (⊤ : add_subgroup V₁))).range = f.range | by simpa [comp_range, incl_range, ← add_monoid_hom.range_eq_map] | lemma | normed_add_group_hom.range_comp_incl_top | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_noninc (f : normed_add_group_hom V W) : Prop | ∀ v, ‖f v‖ ≤ ‖v‖ | def | normed_add_group_hom.norm_noninc | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | A `normed_add_group_hom` is *norm-nonincreasing* if `‖f v‖ ≤ ‖v‖` for all `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_noninc_iff_norm_le_one : f.norm_noninc ↔ ‖f‖ ≤ 1 | begin
refine ⟨λ h, _, λ h, λ v, _⟩,
{ refine op_norm_le_bound _ (zero_le_one) (λ v, _),
simpa [one_mul] using h v },
{ simpa using le_of_op_norm_le f h v }
end | lemma | normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : (0 : normed_add_group_hom V₁ V₂).norm_noninc | λ v, by simp | lemma | normed_add_group_hom.norm_noninc.zero | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : (id V).norm_noninc | λ v, le_rfl | lemma | normed_add_group_hom.norm_noninc.id | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : normed_add_group_hom V₂ V₃} {f : normed_add_group_hom V₁ V₂}
(hg : g.norm_noninc) (hf : f.norm_noninc) :
(g.comp f).norm_noninc | λ v, (hg (f v)).trans (hf v) | lemma | normed_add_group_hom.norm_noninc.comp | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_iff {f : normed_add_group_hom V₁ V₂} : (-f).norm_noninc ↔ f.norm_noninc | ⟨λ h x, by { simpa using h x }, λ h x, (norm_neg (f x)).le.trans (h x)⟩ | lemma | normed_add_group_hom.norm_noninc.neg_iff | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_of_isometry {f : normed_add_group_hom V W} (hf : isometry f) (v : V) :
‖f v‖ = ‖v‖ | (add_monoid_hom_class.isometry_iff_norm f).mp hf v | lemma | normed_add_group_hom.norm_eq_of_isometry | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"isometry",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_id : @isometry V V _ _ (id V) | isometry_id | lemma | normed_add_group_hom.isometry_id | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"isometry",
"isometry_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_comp {g : normed_add_group_hom V₂ V₃} {f : normed_add_group_hom V₁ V₂}
(hg : isometry g) (hf : isometry f) :
isometry (g.comp f) | hg.comp hf | lemma | normed_add_group_hom.isometry_comp | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"isometry",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_noninc_of_isometry (hf : isometry f) : f.norm_noninc | λ v, le_of_eq $ norm_eq_of_isometry hf v | lemma | normed_add_group_hom.norm_noninc_of_isometry | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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