Theorems · Theorem · functional analysis
norm_inner_le_norm
- #78 of the 100 theorems: The Cauchy-Schwarz Inequality
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(x y : E), ‖inner 𝕜 x y‖ ≤ ‖x‖ * ‖y‖Cauchy–Schwarz inequality with norm
- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 170 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.coeproof · cited by 62,936
- Realstatement and proof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Inner.innerstatement and proof · cited by 1,089
- Real.sqrtproof · cited by 545
- RCLike.reproof · cited by 319
- PreInnerProductSpace.Coreproof · cited by 46
- norm_eq_sqrt_re_innerproof · cited by 5
- PreInnerProductSpace.toCoreproof · cited by 2
Cited by14
Results whose statement or proof uses this declaration.
- abs_real_inner_le_normproof · cited by 7
- isBoundedBilinearMap_innerproof · cited by 6
- innerSL_apply_normproof · cited by 5
- MeasureTheory.MemLp.const_innerproof · cited by 4
- ContinuousLinearMap.rayleighQuotient_le_normproof · cited by 2
- inner_eq_zero_of_leftproof · cited by 2
- MeasureTheory.MemLp.inner_constproof · cited by 1
- nnnorm_inner_le_nnnormproof · cited by 1
- lp.summable_innerproof · cited by 1
- ContinuousLinearMap.antilipschitz_of_forall_le_inner_mapproof · cited by 1
- re_inner_le_normproof · cited by 1
- MeasureTheory.isTightMeasureSet_iff_inner_tendstoproof · cited by 1