Theorems · Theorem · global analysis

HasGradientWithinAt.hasFDerivWithinAt

∀ {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} {s : Set F},
  HasGradientWithinAt f f' s x → HasFDerivWithinAt f ((InnerProductSpace.toDual 𝕜 F) f') s x

Alias of the forward direction of hasGradientWithinAt_iff_hasFDerivWithinAt.

Defined in: Mathlib.Analysis.Calculus.Gradient.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace CompleteSpace

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Cites14

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement · cited by 62,936
Setstatement and proof · cited by 53,352
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
HasFDerivWithinAtstatement · cited by 356
InnerProductSpace.toDualstatement · cited by 45

Cited by2

Results whose statement or proof uses this declaration.

HasGradientWithinAt.differentiableWithinAtproof · cited by 0
HasGradientWithinAt.fderivWithin_applyproof · cited by 0