Theorems · Definition · functional analysis

IsBoundedBilinearMap.toContinuousLinearMap

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_2} →
      [inst_1 : SeminormedAddCommGroup E] →
        [inst_2 : NormedSpace 𝕜 E] →
          {F : Type u_3} →
            [inst_3 : SeminormedAddCommGroup F] →
              [inst_4 : NormedSpace 𝕜 F] →
                {G : Type u_4} →
                  [inst_5 : SeminormedAddCommGroup G] →
                    [inst_6 : NormedSpace 𝕜 G] → {f : E × F → G} → IsBoundedBilinearMap 𝕜 f → E →L[𝕜] F →L[𝕜] G

A bounded bilinear map f : E × F → G defines a continuous linear map f : E →L[𝕜] F →L[𝕜] G.

Defined in

Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Cited by

8 results in Mathlib

Foundations

Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpace