Theorems · Theorem · functional analysis
ContinuousLinearMap.unit_le_opNorm
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : SeminormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : NontriviallyNormedField 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)
(x : E), ‖x‖ ≤ 1 → ‖f x‖ ≤ ‖f‖The image of the unit ball under a continuous linear map is bounded.
- Defined in
- Mathlib.Analysis.Normed.Operator.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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 · cited by 25,697
- 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
- mul_oneproof · cited by 3,885
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- RingHomIsometricstatement and proof · cited by 282
- ContinuousLinearMap.le_opNorm_of_leproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- Complex.reCLM_normproof · cited by 2
- Complex.imCLM_normproof · cited by 2
- ContinuousLinearMap.sSup_unitClosedBall_eq_nnnormproof · cited by 1
- Unitization.norm_splitMul_snd_sqproof · cited by 0