Theorems · Definition · functional analysis

LinearMap.mkContinuous

{𝕜 : Type u_1} →
  {𝕜₂ : Type u_2} →
    {E : Type u_3} →
      {F : Type u_4} →
        [inst : Ring 𝕜] →
          [inst_1 : Ring 𝕜₂] →
            [inst_2 : SeminormedAddCommGroup E] →
              [inst_3 : SeminormedAddCommGroup F] →
                [inst_4 : Module 𝕜 E] →
                  [inst_5 : Module 𝕜₂ F] →
                    {σ : 𝕜 →+* 𝕜₂} → (f : E →ₛₗ[σ] F) → (C : ℝ) → (∀ (x : E), ‖f x‖ ≤ C * ‖x‖) → E →SL[σ] F

Construct a continuous linear map from a linear map and a bound on this linear map. The fact that the norm of the continuous linear map is then controlled is given in LinearMap.mkContinuous_norm_le.

Defined in

Mathlib.Analysis.Normed.Operator.ContinuousLinearMap

Cited by

9 results in Mathlib

Foundations

Depth 154 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingRingSeminormedAddCommGroupSeminormedAddCommGroupModuleModule