Theorems · Theorem · functional analysis
InnerProductSpace.norm_sq_eq_re_inner
∀ {𝕜 : Type u_4} {E : Type u_5} {inst : RCLike 𝕜} {inst_1 : SeminormedAddCommGroup E} [self : InnerProductSpace 𝕜 E]
(x : E), ‖x‖ ^ 2 = RCLike.re (inner 𝕜 x x)The inner product induces the norm.
- Defined in
- Mathlib.Analysis.InnerProductSpace.Defs
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- InnerProductSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Norm.normstatement · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- AddMonoidHomstatement · cited by 3,230
- RCLikestatement and proof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Inner.innerstatement · cited by 1,089
- AddMonoid.toZerostatement · cited by 325
- RCLike.restatement · cited by 319
Cited by8
Results whose statement or proof uses this declaration.
- inner_self_eq_norm_sq_to_Kproof · cited by 72
- orthonormal_iff_iteproof · cited by 15
- norm_eq_sqrt_re_innerproof · cited by 5
- InnerProductSpace.rclikeToRealproof · cited by 5
- norm_inner_eq_norm_tfaeproof · cited by 3
- Orthonormal.sum_inner_products_leproof · cited by 2
- ContinuousLinearMap.opNorm_le_of_re_inner_leproof · cited by 1
- InnerProductGeometry.norm_ofLp_crossProductproof · cited by 1