Theorems · Theorem · functional analysis
norm_smul_inv_norm
∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {x : E},
x ≠ 0 → ‖(↑‖x‖)⁻¹ • x‖ = 1Lemma to normalize a vector in a normed space E over either ℂ or ℝ to unit length.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Norm.normstatement and proof · cited by 5,413
- one_mulproof · cited by 2,841
- RCLikestatement and proof · cited by 2,829
- one_ne_zeroproof · cited by 885
- div_oneproof · cited by 629
- RCLike.ofRealstatement and proof · cited by 350
- norm_smulproof · cited by 242
- norm_invproof · cited by 126
- norm_normproof · cited by 113
Cited by5
Results whose statement or proof uses this declaration.
- maximal_orthonormal_iff_orthogonalComplement_eq_botproof · cited by 3
- InnerProductSpace.gramSchmidtNormed_unit_length'proof · cited by 1
- InnerProductSpace.gramSchmidtNormed_unit_length_coeproof · cited by 1
- antilipschitz_of_bound_of_norm_oneproof · cited by 0
- riesz_lemma_of_lt_oneproof · cited by 0