Theorems · Theorem · functional analysis
PreInnerProductSpace.Core.re_inner_nonneg
∀ {𝕜 : Type u_4} {F : Type u_5} [inst : RCLike 𝕜] [inst_1 : AddCommGroup F] [inst_2 : Module 𝕜 F]
(self : PreInnerProductSpace.Core 𝕜 F) (x : F), 0 ≤ RCLike.re (inner 𝕜 x x)The inner product is positive (semi)definite.
- Defined in
- Mathlib.Analysis.InnerProductSpace.Defs
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RCLikeAddCommGroupModule
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
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- AddMonoidHomstatement · cited by 3,230
- RCLikestatement and proof · cited by 2,829
- Inner.innerstatement · cited by 1,089
- AddMonoid.toZerostatement · cited by 325
- RCLike.restatement · cited by 319
- PreInnerProductSpace.Corestatement and proof · cited by 46
- PreInnerProductSpace.Core.toInnerstatement · cited by 9
Cited by2
Results whose statement or proof uses this declaration.
- inner_self_nonnegproof · cited by 6
- InnerProductSpace.Core.inner_self_nonnegproof · cited by 4