Theorems · Theorem · functional analysis
ContinuousLinearMap.exists_mul_lt_apply_of_lt_opNNNorm
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
[inst_1 : NontriviallyNormedField 𝕜₂] [inst_2 : SeminormedAddCommGroup E] [inst_3 : SeminormedAddCommGroup F]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_6 : RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F) {r : NNReal}, r < ‖f‖₊ → ∃ x, r * ‖x‖₊ < ‖f x‖₊- Defined in
- Mathlib.Analysis.Normed.Operator.NNNorm
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 173 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.
- DFunLike.coestatement and proof · cited by 62,936
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Set.ofPredproof · cited by 6,101
- ContinuousLinearMapstatement and proof · cited by 5,352
- NNRealstatement and proof · cited by 4,310
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- NNNorm.nnnormstatement and proof · cited by 952
- RingHomIsometricstatement and proof · cited by 282
- OrderBot.bddBelowproof · cited by 31
- notMem_of_lt_csInfproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.exists_lt_apply_of_lt_opNNNormproof · cited by 3
- ContinuousLinearMap.exists_mul_lt_of_lt_opNormproof · cited by 0