Theorems · Theorem · operator theory

isCompactOperator_iff_image_closedBall_subset_compact

∀ {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [inst : NontriviallyNormedField 𝕜₁] [inst_1 : SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂}
  {M₁ : Type u_3} {M₂ : Type u_4} [inst_2 : SeminormedAddCommGroup M₁] [inst_3 : TopologicalSpace M₂]
  [inst_4 : AddCommMonoid M₂] [inst_5 : NormedSpace 𝕜₁ M₁] [inst_6 : Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂]
  (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ}, 0 < r → (IsCompactOperator ⇑f ↔ ∃ K, IsCompact K ∧ ⇑f '' Metric.closedBall 0 r ⊆ K)

Defined in

Mathlib.Analysis.Normed.Operator.Compact.Basic

Cited by

0 results in Mathlib

Foundations

Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedRingSeminormedAddCommGroupTopologicalSpaceAddCommMonoidNormedSpaceModuleContinuousConstSMul