Rays in a real normed vector space #

In this file we prove some lemmas about the SameRay predicate in case of a real normed space. In this case, for two vectors x y in the same ray, the norm of their sum is equal to the sum of their norms and ‖y‖ • x = ‖x‖ • y.

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theorem SameRay.norm_add {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x : E} {y : E} (h : SameRay ℝ x y) :

‖x + y‖ = ‖x‖ + ‖y‖

If x and y are on the same ray, then the triangle inequality becomes the equality: the norm of x + y is the sum of the norms of x and y. The converse is true for a strictly convex space.

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theorem SameRay.norm_sub {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x : E} {y : E} (h : SameRay ℝ x y) :

‖x - y‖ = |‖x‖ - ‖y‖|

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theorem SameRay.norm_smul_eq {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x : E} {y : E} (h : SameRay ℝ x y) :

‖x‖ • y = ‖y‖ • x

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theorem norm_injOn_ray_left {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} (hx : x ≠ 0) :

Set.InjOn norm {y | SameRay ℝ x y}

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theorem norm_injOn_ray_right {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {y : F} (hy : y ≠ 0) :

Set.InjOn norm {x | SameRay ℝ x y}

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theorem sameRay_iff_norm_smul_eq {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} :

SameRay ℝ x y ↔ ‖x‖ • y = ‖y‖ • x

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theorem sameRay_iff_inv_norm_smul_eq_of_ne {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} (hx : x ≠ 0) (hy : y ≠ 0) :

SameRay ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y

Two nonzero vectors x y in a real normed space are on the same ray if and only if the unit vectors ‖x‖⁻¹ • x and ‖y‖⁻¹ • y are equal.

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theorem SameRay.inv_norm_smul_eq {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} (hx : x ≠ 0) (hy : y ≠ 0) :

SameRay ℝ x y → ‖x‖⁻¹ • x = ‖y‖⁻¹ • y

Alias of the forward direction of sameRay_iff_inv_norm_smul_eq_of_ne.

Two nonzero vectors x y in a real normed space are on the same ray if and only if the unit vectors ‖x‖⁻¹ • x and ‖y‖⁻¹ • y are equal.

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theorem sameRay_iff_inv_norm_smul_eq {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} :

SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y

Two vectors x y in a real normed space are on the ray if and only if one of them is zero or the unit vectors ‖x‖⁻¹ • x and ‖y‖⁻¹ • y are equal.

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theorem sameRay_iff_of_norm_eq {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} (h : ‖x‖ = ‖y‖) :

SameRay ℝ x y ↔ x = y

Two vectors of the same norm are on the same ray if and only if they are equal.

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theorem not_sameRay_iff_of_norm_eq {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} (h : ‖x‖ = ‖y‖) :

¬SameRay ℝ x y ↔ x ≠ y

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theorem SameRay.eq_of_norm_eq {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} (h : SameRay ℝ x y) (hn : ‖x‖ = ‖y‖) :

x = y

If two points on the same ray have the same norm, then they are equal.

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theorem SameRay.norm_eq_iff {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {y : F} (h : SameRay ℝ x y) :

‖x‖ = ‖y‖ ↔ x = y

The norms of two vectors on the same ray are equal if and only if they are equal.