Theorems · Theorem · functional analysis
ContinuousLinearMap.exists_lt_apply_of_lt_opNorm
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : NormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : DenselyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
{r : ℝ}, r < ‖f‖ → ∃ x, ‖x‖ < 1 ∧ r < ‖f x‖- Defined in
- Mathlib.Analysis.Normed.Operator.NNNorm
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normstatement and proof · cited by 5,413
- ContinuousLinearMapstatement and proof · cited by 5,352
- NNRealproof · cited by 4,310
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- map_zeroproof · cited by 1,614
- NNReal.toRealproof · cited by 1,260
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