Seminorms and norms on rings

This file defines seminorms and norms on rings. These definitions are useful when one needs to consider multiple (semi)norms on a given ring.

Main declarations

For a ring R:

• RingSeminorm: A seminorm on a ring R is a function f : R → ℝ that preserves zero, takes nonnegative values, is subadditive and submultiplicative and such that f (-x) = f x for all x ∈ R.
• RingNorm: A seminorm f is a norm if f x = 0 if and only if x = 0.
• MulRingSeminorm: A multiplicative seminorm on a ring R is a ring seminorm that preserves multiplication.
• MulRingNorm: A multiplicative norm on a ring R is a ring norm that preserves multiplication.

Notes

The corresponding hom classes are defined in Mathlib.Analysis.Order.Hom.Basic to be used by absolute values.

References

• [S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis][bosch-guntzer-remmert]

Tags

ring_seminorm, ring_norm

structure RingSeminorm (R : Type u_4) [NonUnitalNonAssocRing R] extends AddGroupSeminorm : Type u_4

A seminorm on a ring R is a function f : R → ℝ that preserves zero, takes nonnegative values, is subadditive and submultiplicative and such that f (-x) = f x for all x ∈ R.

• toFun : R → ℝ
• map_zero' : self.toFun 0 = 0
• add_le' : ∀ (r s : R), self.toFun (r + s) ≤ self.toFun r + self.toFun s
• neg' : ∀ (r : R), self.toFun (-r) = self.toFun r
• mul_le' : ∀ (x y : R), self.toFun (x * y) ≤ self.toFun x * self.toFun y

  The property of a RingSeminorm that for all x and y in the ring, the norm of x * y is less than the norm of x times the norm of y.

theorem RingSeminorm.mul_le' {R : Type u_4} [NonUnitalNonAssocRing R] (self : RingSeminorm R) (x : R) (y : R) : self.toFun (x * y) ≤ self.toFun x * self.toFun y

The property of a RingSeminorm that for all x and y in the ring, the norm of x * y is less than the norm of x times the norm of y.

structure RingNorm (R : Type u_4) [NonUnitalNonAssocRing R] extends RingSeminorm : Type u_4

A function f : R → ℝ is a norm on a (nonunital) ring if it is a seminorm and f x = 0 implies x = 0.

• toFun : R → ℝ
• map_zero' : self.toFun 0 = 0
• add_le' : ∀ (r s : R), self.toFun (r + s) ≤ self.toFun r + self.toFun s
• neg' : ∀ (r : R), self.toFun (-r) = self.toFun r
• mul_le' : ∀ (x y : R), self.toFun (x * y) ≤ self.toFun x * self.toFun y
• eq_zero_of_map_eq_zero' : ∀ (x : R), self.toFun x = 0 → x = 0

  If the image under the seminorm is zero, then the argument is zero.

structure MulRingSeminorm (R : Type u_4) [NonAssocRing R] extends AddGroupSeminorm : Type u_4

A multiplicative seminorm on a ring R is a function f : R → ℝ that preserves zero and multiplication, takes nonnegative values, is subadditive and such that f (-x) = f x for all x.

• toFun : R → ℝ
• map_zero' : self.toFun 0 = 0
• add_le' : ∀ (r s : R), self.toFun (r + s) ≤ self.toFun r + self.toFun s
• neg' : ∀ (r : R), self.toFun (-r) = self.toFun r
• map_one' : self.toFun 1 = 1

  The proposition that the function preserves 1

• map_mul' : ∀ (x y : R), self.toFun (x * y) = self.toFun x * self.toFun y

  The proposition that the function preserves multiplication

structure MulRingNorm (R : Type u_4) [NonAssocRing R] extends MulRingSeminorm : Type u_4

A multiplicative norm on a ring R is a multiplicative ring seminorm such that f x = 0 implies x = 0.

• toFun : R → ℝ
• map_zero' : self.toFun 0 = 0
• add_le' : ∀ (r s : R), self.toFun (r + s) ≤ self.toFun r + self.toFun s
• neg' : ∀ (r : R), self.toFun (-r) = self.toFun r
• map_one' : self.toFun 1 = 1
• map_mul' : ∀ (x y : R), self.toFun (x * y) = self.toFun x * self.toFun y
• eq_zero_of_map_eq_zero' : ∀ (x : R), self.toFun x = 0 → x = 0

  If the image under the seminorm is zero, then the argument is zero.

instance RingSeminorm.funLike {R : Type u_2} [NonUnitalRing R] : FunLike (RingSeminorm R) R ℝ

Equations

• RingSeminorm.funLike = { coe := fun (f : RingSeminorm R) => f.toFun, coe_injective' := ⋯ }

instance RingSeminorm.ringSeminormClass {R : Type u_2} [NonUnitalRing R] : RingSeminormClass (RingSeminorm R) R ℝ

Equations

• ⋯ = ⋯

@[simp]

theorem RingSeminorm.toFun_eq_coe {R : Type u_2} [NonUnitalRing R] (p : RingSeminorm R) : ⇑p.toAddGroupSeminorm = ⇑p

theorem RingSeminorm.ext {R : Type u_2} [NonUnitalRing R] {p : RingSeminorm R} {q : RingSeminorm R} : (∀ (x : R), p x = q x) → p = q

instance RingSeminorm.instZero {R : Type u_2} [NonUnitalRing R] : Zero (RingSeminorm R)

Equations

• RingSeminorm.instZero = { zero := let __src := Zero.zero; { toAddGroupSeminorm := __src, mul_le' := ⋯ } }

theorem RingSeminorm.eq_zero_iff {R : Type u_2} [NonUnitalRing R] {p : RingSeminorm R} : p = 0 ↔ ∀ (x : R), p x = 0

theorem RingSeminorm.ne_zero_iff {R : Type u_2} [NonUnitalRing R] {p : RingSeminorm R} : p ≠ 0 ↔ ∃ (x : R), p x ≠ 0

instance RingSeminorm.instInhabited {R : Type u_2} [NonUnitalRing R] : Inhabited (RingSeminorm R)

Equations

• RingSeminorm.instInhabited = { default := 0 }

instance RingSeminorm.instOneOfDecidableEq {R : Type u_2} [NonUnitalRing R] [DecidableEq R] : One (RingSeminorm R)

The trivial seminorm on a ring R is the RingSeminorm taking value 0 at 0 and 1 at every other element.

Equations

• RingSeminorm.instOneOfDecidableEq = { one := let __src := 1; { toAddGroupSeminorm := __src, mul_le' := ⋯ } }

@[simp]

theorem RingSeminorm.apply_one {R : Type u_2} [NonUnitalRing R] [DecidableEq R] (x : R) : 1 x = if x = 0 then 0 else 1

theorem RingSeminorm.seminorm_one_eq_one_iff_ne_zero {R : Type u_2} [Ring R] (p : RingSeminorm R) (hp : p 1 ≤ 1) : p 1 = 1 ↔ p ≠ 0

def normRingSeminorm (R : Type u_4) [NonUnitalSeminormedRing R] : RingSeminorm R

The norm of a NonUnitalSeminormedRing as a RingSeminorm.

Equations

• normRingSeminorm R = let __src := normAddGroupSeminorm R; { toFun := norm, map_zero' := ⋯, add_le' := ⋯, neg' := ⋯, mul_le' := ⋯ }

instance RingNorm.funLike {R : Type u_2} [NonUnitalRing R] : FunLike (RingNorm R) R ℝ

Equations

• RingNorm.funLike = { coe := fun (f : RingNorm R) => f.toFun, coe_injective' := ⋯ }

instance RingNorm.ringNormClass {R : Type u_2} [NonUnitalRing R] : RingNormClass (RingNorm R) R ℝ

Equations

• ⋯ = ⋯

theorem RingNorm.toFun_eq_coe {R : Type u_2} [NonUnitalRing R] (p : RingNorm R) : p.toFun = ⇑p

theorem RingNorm.ext {R : Type u_2} [NonUnitalRing R] {p : RingNorm R} {q : RingNorm R} : (∀ (x : R), p x = q x) → p = q

instance RingNorm.instOneOfDecidableEq (R : Type u_2) [NonUnitalRing R] [DecidableEq R] : One (RingNorm R)

The trivial norm on a ring R is the RingNorm taking value 0 at 0 and 1 at every other element.

Equations

• RingNorm.instOneOfDecidableEq R = { one := let __src := 1; let __src_1 := 1; { toRingSeminorm := __src, eq_zero_of_map_eq_zero' := ⋯ } }

@[simp]

theorem RingNorm.apply_one (R : Type u_2) [NonUnitalRing R] [DecidableEq R] (x : R) : 1 x = if x = 0 then 0 else 1

instance RingNorm.instInhabitedOfDecidableEq (R : Type u_2) [NonUnitalRing R] [DecidableEq R] : Inhabited (RingNorm R)

Equations

• RingNorm.instInhabitedOfDecidableEq R = { default := 1 }

instance MulRingSeminorm.funLike {R : Type u_2} [NonAssocRing R] : FunLike (MulRingSeminorm R) R ℝ

Equations

• MulRingSeminorm.funLike = { coe := fun (f : MulRingSeminorm R) => f.toFun, coe_injective' := ⋯ }

instance MulRingSeminorm.mulRingSeminormClass {R : Type u_2} [NonAssocRing R] : MulRingSeminormClass (MulRingSeminorm R) R ℝ

Equations

• ⋯ = ⋯

@[simp]

theorem MulRingSeminorm.toFun_eq_coe {R : Type u_2} [NonAssocRing R] (p : MulRingSeminorm R) : ⇑p.toAddGroupSeminorm = ⇑p

theorem MulRingSeminorm.ext {R : Type u_2} [NonAssocRing R] {p : MulRingSeminorm R} {q : MulRingSeminorm R} : (∀ (x : R), p x = q x) → p = q

instance MulRingSeminorm.instOne {R : Type u_2} [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] : One (MulRingSeminorm R)

The trivial seminorm on a ring R is the MulRingSeminorm taking value 0 at 0 and 1 at every other element.

Equations

• MulRingSeminorm.instOne = { one := let __src := 1; { toAddGroupSeminorm := __src, map_one' := ⋯, map_mul' := ⋯ } }

@[simp]

theorem MulRingSeminorm.apply_one {R : Type u_2} [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] (x : R) : 1 x = if x = 0 then 0 else 1

instance MulRingSeminorm.instInhabited {R : Type u_2} [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] : Inhabited (MulRingSeminorm R)

Equations

• MulRingSeminorm.instInhabited = { default := 1 }

instance MulRingNorm.funLike {R : Type u_2} [NonAssocRing R] : FunLike (MulRingNorm R) R ℝ

Equations

• MulRingNorm.funLike = { coe := fun (f : MulRingNorm R) => f.toFun, coe_injective' := ⋯ }

instance MulRingNorm.mulRingNormClass {R : Type u_2} [NonAssocRing R] : MulRingNormClass (MulRingNorm R) R ℝ

Equations

• ⋯ = ⋯

theorem MulRingNorm.toFun_eq_coe {R : Type u_2} [NonAssocRing R] (p : MulRingNorm R) : p.toFun = ⇑p

theorem MulRingNorm.ext {R : Type u_2} [NonAssocRing R] {p : MulRingNorm R} {q : MulRingNorm R} : (∀ (x : R), p x = q x) → p = q

instance MulRingNorm.instOne (R : Type u_2) [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] : One (MulRingNorm R)

The trivial norm on a ring R is the MulRingNorm taking value 0 at 0 and 1 at every other element.

Equations

• MulRingNorm.instOne R = { one := let __src := 1; let __src_1 := 1; { toMulRingSeminorm := __src, eq_zero_of_map_eq_zero' := ⋯ } }

@[simp]

theorem MulRingNorm.apply_one (R : Type u_2) [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] (x : R) : 1 x = if x = 0 then 0 else 1

instance MulRingNorm.instInhabited (R : Type u_2) [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] : Inhabited (MulRingNorm R)

Equations

• MulRingNorm.instInhabited R = { default := 1 }

def MulRingNorm.equiv {R : Type u_4} [Ring R] (f : MulRingNorm R) (g : MulRingNorm R) : Prop

Two multiplicative ring norms f, g on R are equivalent if there exists a positive constant c such that for all x ∈ R, (f x)^c = g x.

Equations

• f.equiv g = ∃ (c : ℝ), 0 < c ∧ (fun (x : R) => f x ^ c) = ⇑g

theorem MulRingNorm.equiv_refl {R : Type u_4} [Ring R] (f : MulRingNorm R) : f.equiv f

Equivalence of multiplicative ring norms is reflexive.

theorem MulRingNorm.equiv_symm {R : Type u_4} [Ring R] {f : MulRingNorm R} {g : MulRingNorm R} (hfg : f.equiv g) : g.equiv f

Equivalence of multiplicative ring norms is symmetric.

theorem MulRingNorm.equiv_trans {R : Type u_4} [Ring R] {f : MulRingNorm R} {g : MulRingNorm R} {k : MulRingNorm R} (hfg : f.equiv g) (hgk : g.equiv k) : f.equiv k

Equivalence of multiplicative ring norms is transitive.

def RingSeminorm.toRingNorm {K : Type u_4} [Field K] (f : RingSeminorm K) (hnt : f ≠ 0) : RingNorm K

A nonzero ring seminorm on a field K is a ring norm.

Equations

• f.toRingNorm hnt = { toRingSeminorm := f, eq_zero_of_map_eq_zero' := ⋯ }

@[simp]

theorem normRingNorm_toFun (R : Type u_4) [NonUnitalNormedRing R] : ∀ (a : R), (normRingNorm R).toFun a = ‖a‖

def normRingNorm (R : Type u_4) [NonUnitalNormedRing R] : RingNorm R

The norm of a NonUnitalNormedRing as a RingNorm.

Equations

• normRingNorm R = let __src := normAddGroupNorm R; let __src_1 := normRingSeminorm R; { toAddGroupSeminorm := __src.toAddGroupSeminorm, mul_le' := ⋯, eq_zero_of_map_eq_zero' := ⋯ }

theorem MulRingNorm_nat_le_nat {R : Type u_4} [Ring R] (n : ℕ) (f : MulRingNorm R) : f ↑n ≤ ↑n

A multiplicative ring norm satisfies f n ≤ n for every n : ℕ.

theorem MulRingNorm.apply_natAbs_eq {R : Type u_4} [Ring R] (x : ℤ) (f : MulRingNorm R) : f ↑x.natAbs = f ↑x

A multiplicative norm composed with the absolute value on integers equals the norm itself.