Theorems · Inductive type · functional analysis

HasFDerivAtFilter

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_2} →
      [inst_1 : AddCommGroup E] →
        [inst_2 : Module 𝕜 E] →
          [inst_3 : TopologicalSpace E] →
            {F : Type u_3} →
              [inst_4 : AddCommGroup F] →
                [inst_5 : Module 𝕜 F] → [inst_6 : TopologicalSpace F] → (E → F) → (E →L[𝕜] F) → Filter (E × E) → Prop

A function f has the continuous linear map f' as derivative along the filter L if f x₁ = f x₂ + f' (x₁ - x₂) + o (x₁ - x₂) when x = (x₁, x₂) converges along the filter L. This definition is designed to be specialized - for L = 𝓝 (x, x) (in HasStrictFDerivAt), giving rise to the derivative in the sense of strict differentiability; - for L = 𝓝 x ×ˢ pure x (in HasFDerivAt), giving rise to the usual notion of Fréchet derivative; - for L = 𝓝[s] x ×ˢ pure x (in HasFDerivWithinAt), giving rise to the notion of Fréchet derivative along the set s.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Defs

Cited by

81 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpace