Theorems · Theorem · field theory
NormedField.exists_norm_lt_one
∀ (α : Type u_2) [inst : NontriviallyNormedField α], ∃ x, 0 < ‖x‖ ∧ ‖x‖ < 1
- Defined in
- Mathlib.Analysis.Normed.Field.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NontriviallyNormedField
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normstatement · cited by 5,413
- one_posproof · cited by 102
- NormedField.exists_norm_ltproof · cited by 11
Cited by6
Results whose statement or proof uses this declaration.
- Valued.integer.exists_norm_coe_lt_oneproof · cited by 2
- ContinuousSMul.topology_eq_of_nhds_inf_principal_eqproof · cited by 1
- LinearMap.continuousAt_zero_of_locally_boundedproof · cited by 1
- continuum_le_cardinal_of_nontriviallyNormedFieldproof · cited by 1
- NormedField.exists_nnnorm_lt_oneproof · cited by 0
- perfectSpace_of_moduleproof · cited by 0