Theorems · Theorem · global analysis
hasGradientAt_iff_hasFDerivAt
∀ {𝕜 : Type u_1} {F : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup F] [inst_2 : InnerProductSpace 𝕜 F]
[inst_3 : CompleteSpace F] {f : F → 𝕜} {f' x : F},
HasGradientAt f f' x ↔ HasFDerivAt f ((InnerProductSpace.toDual 𝕜 F) f') x- Defined in
- Mathlib.Analysis.Calculus.Gradient.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- CompleteSpacestatement and proof · cited by 2,532
- LinearIsometryEquivstatement · cited by 748
- starRingEndstatement · cited by 671
- StrongDualstatement · cited by 459
- HasFDerivAtstatement · cited by 350
- InnerProductSpace.toDualstatement · cited by 45
- HasGradientAtstatement · cited by 24
Cited by5
Results whose statement or proof uses this declaration.
- HasGradientAt.hasFDerivAtproof · cited by 3
- HasGradientAt.hasDerivAtproof · cited by 2
- hasFDerivAt_iff_hasGradientAtproof · cited by 1
- HasDerivAt.hasGradientAtproof · cited by 1
- hasGradientWithinAt_univproof · cited by 0