Documentation

Mathlib.Analysis.Normed.Operator.Compact

Compact operators #

In this file we define compact linear operators between two topological vector spaces (TVS).

Main definitions #

IsCompactOperator : predicate for compact operators

Main statements #

isCompactOperator_iff_isCompact_closure_image_ball : the usual characterization of compact operators from a normed space to a T2 TVS.
IsCompactOperator.comp_clm : precomposing a compact operator by a continuous linear map gives a compact operator
IsCompactOperator.clm_comp : postcomposing a compact operator by a continuous linear map gives a compact operator
IsCompactOperator.continuous : compact operators are automatically continuous
isClosed_setOf_isCompactOperator : the set of compact operators is closed for the operator norm

Implementation details #

We define IsCompactOperator as a predicate, because the space of compact operators inherits all of its structure from the space of continuous linear maps (e.g we want to have the usual operator norm on compact operators).

The two natural options then would be to make it a predicate over linear maps or continuous linear maps. Instead we define it as a predicate over bare functions, although it really only makes sense for linear functions, because Lean is really good at finding coercions to bare functions (whereas coercing from continuous linear maps to linear maps often needs type ascriptions).

References #

[N. Bourbaki, Théories Spectrales, Chapitre 3][bourbaki2023]

Tags #

Compact operator

def IsCompactOperator {M₁ : Type u_1} {M₂ : Type u_2} [Zero M₁] [TopologicalSpace M₁] [TopologicalSpace M₂] (f : M₁ → M₂) :

A compact operator between two topological vector spaces. This definition is usually given as "there exists a neighborhood of zero whose image is contained in a compact set", but we choose a definition which involves fewer existential quantifiers and replaces images with preimages.

We prove the equivalence in isCompactOperator_iff_exists_mem_nhds_image_subset_compact.

Equations

Instances For

theorem isCompactOperator_zero {M₁ : Type u_1} {M₂ : Type u_2} [Zero M₁] [TopologicalSpace M₁] [TopologicalSpace M₂] [Zero M₂] :

IsCompactOperator 0

theorem isCompactOperator_iff_exists_mem_nhds_image_subset_compact {M₁ : Type u_2} {M₂ : Type u_3} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] (f : M₁ → M₂) :

IsCompactOperator f ↔ ∃ V ∈ nhds 0, ∃ (K : Set M₂), IsCompact K ∧ f '' V ⊆ K

theorem isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image {M₁ : Type u_2} {M₂ : Type u_3} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [T2Space M₂] (f : M₁ → M₂) :

IsCompactOperator f ↔ ∃ V ∈ nhds 0, IsCompact (closure (f '' V))

theorem IsCompactOperator.image_subset_compact_of_isVonNBounded {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsVonNBounded 𝕜₁ S) :

∃ (K : Set M₂), IsCompact K ∧ ⇑f '' S ⊆ K

theorem IsCompactOperator.isCompact_closure_image_of_isVonNBounded {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsVonNBounded 𝕜₁ S) :

IsCompact (closure (⇑f '' S))

theorem IsCompactOperator.image_subset_compact_of_bounded {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsBounded S) :

∃ (K : Set M₂), IsCompact K ∧ ⇑f '' S ⊆ K

theorem IsCompactOperator.isCompact_closure_image_of_bounded {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsBounded S) :

IsCompact (closure (⇑f '' S))

theorem IsCompactOperator.image_ball_subset_compact {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ) :

∃ (K : Set M₂), IsCompact K ∧ ⇑f '' Metric.ball 0 r ⊆ K

theorem IsCompactOperator.image_closedBall_subset_compact {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ) :

∃ (K : Set M₂), IsCompact K ∧ ⇑f '' Metric.closedBall 0 r ⊆ K

theorem IsCompactOperator.isCompact_closure_image_ball {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ) :

IsCompact (closure (⇑f '' Metric.ball 0 r))

theorem IsCompactOperator.isCompact_closure_image_closedBall {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ) :

IsCompact (closure (⇑f '' Metric.closedBall 0 r))

theorem isCompactOperator_iff_image_ball_subset_compact {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :

IsCompactOperator ⇑f ↔ ∃ (K : Set M₂), IsCompact K ∧ ⇑f '' Metric.ball 0 r ⊆ K

theorem isCompactOperator_iff_image_closedBall_subset_compact {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :

IsCompactOperator ⇑f ↔ ∃ (K : Set M₂), IsCompact K ∧ ⇑f '' Metric.closedBall 0 r ⊆ K

theorem isCompactOperator_iff_isCompact_closure_image_ball {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :

IsCompactOperator ⇑f ↔ IsCompact (closure (⇑f '' Metric.ball 0 r))

theorem isCompactOperator_iff_isCompact_closure_image_closedBall {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :

IsCompactOperator ⇑f ↔ IsCompact (closure (⇑f '' Metric.closedBall 0 r))

theorem IsCompactOperator.smul {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] {S : Type u_6} [Monoid S] [DistribMulAction S M₂] [ContinuousConstSMul S M₂] {f : M₁ → M₂} (hf : IsCompactOperator f) (c : S) :

IsCompactOperator (c • f)

theorem IsCompactOperator.smul_unit_iff {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] {S : Type u_6} [Monoid S] [DistribMulAction S M₂] [ContinuousConstSMul S M₂] {f : M₁ → M₂} {c : Sˣ} :

IsCompactOperator (c • f) ↔ IsCompactOperator f

theorem IsCompactOperator.smul_isUnit_iff {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] {S : Type u_6} [Monoid S] [DistribMulAction S M₂] [ContinuousConstSMul S M₂] {f : M₁ → M₂} {c : S} (hc : IsUnit c) :

IsCompactOperator (c • f) ↔ IsCompactOperator f

theorem IsCompactOperator.smul_iff {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] {S : Type u_6} [Group S] [DistribMulAction S M₂] [ContinuousConstSMul S M₂] {f : M₁ → M₂} (c : S) :

IsCompactOperator (c • f) ↔ IsCompactOperator f

theorem IsCompactOperator.smul_iff₀ {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] {S : Type u_6} [GroupWithZero S] [DistribMulAction S M₂] [ContinuousConstSMul S M₂] {f : M₁ → M₂} {c : S} (hc : c ≠ 0) :

IsCompactOperator (c • f) ↔ IsCompactOperator f

theorem IsCompactOperator.add {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [ContinuousAdd M₂] {f g : M₁ → M₂} (hf : IsCompactOperator f) (hg : IsCompactOperator g) :

IsCompactOperator (f + g)

theorem IsCompactOperator.neg {M₁ : Type u_3} {M₄ : Type u_5} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₄] [AddCommGroup M₄] [ContinuousNeg M₄] {f : M₁ → M₄} (hf : IsCompactOperator f) :

IsCompactOperator (-f)

theorem IsCompactOperator.sub {M₁ : Type u_3} {M₄ : Type u_5} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₄] [AddCommGroup M₄] [IsTopologicalAddGroup M₄] {f g : M₁ → M₄} (hf : IsCompactOperator f) (hg : IsCompactOperator g) :

IsCompactOperator (f - g)

def compactOperator {R₁ : Type u_1} {R₄ : Type u_2} [Semiring R₁] [CommSemiring R₄] (σ₁₄ : R₁ →+* R₄) (M₁ : Type u_3) (M₄ : Type u_5) [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₄] [AddCommGroup M₄] [Module R₁ M₁] [Module R₄ M₄] [ContinuousConstSMul R₄ M₄] [IsTopologicalAddGroup M₄] :

Submodule R₄ (M₁ →SL[σ₁₄] M₄)

The submodule of compact continuous linear maps.

Equations

Instances For

theorem IsCompactOperator.comp_clm {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [TopologicalSpace M₁] [TopologicalSpace M₂] [TopologicalSpace M₃] [AddCommMonoid M₁] [Module R₁ M₁] [AddCommMonoid M₂] [Module R₂ M₂] {f : M₂ → M₃} (hf : IsCompactOperator f) (g : M₁ →SL[σ₁₂] M₂) :

IsCompactOperator (f ∘ ⇑g)

theorem IsCompactOperator.continuous_comp {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [TopologicalSpace M₁] [TopologicalSpace M₂] [TopologicalSpace M₃] [AddCommMonoid M₁] {f : M₁ → M₂} (hf : IsCompactOperator f) {g : M₂ → M₃} (hg : Continuous g) :

IsCompactOperator (g ∘ f)

theorem IsCompactOperator.clm_comp {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₂] [Semiring R₃] {σ₂₃ : R₂ →+* R₃} {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [TopologicalSpace M₁] [TopologicalSpace M₂] [TopologicalSpace M₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M₃] [Module R₃ M₃] {f : M₁ → M₂} (hf : IsCompactOperator f) (g : M₂ →SL[σ₂₃] M₃) :

IsCompactOperator (⇑g ∘ f)

theorem IsCompactOperator.codRestrict {R₂ : Type u_1} [Semiring R₂] {M₁ : Type u_2} {M₂ : Type u_3} [TopologicalSpace M₁] [TopologicalSpace M₂] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R₂ M₂] {f : M₁ → M₂} (hf : IsCompactOperator f) {V : Submodule R₂ M₂} (hV : ∀ (x : M₁), f x ∈ V) (h_closed : IsClosed ↑V) :

IsCompactOperator (Set.codRestrict f (↑V) hV)

theorem IsCompactOperator.restrict {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_3} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {f : M₁ →ₗ[R₁] M₁} (hf : IsCompactOperator ⇑f) {V : Submodule R₁ M₁} (hV : ∀ v ∈ V, f v ∈ V) (h_closed : IsClosed ↑V) :

IsCompactOperator ⇑(f.restrict hV)

If a compact operator preserves a closed submodule, its restriction to that submodule is compact.

Note that, following mathlib's convention in linear algebra, restrict designates the restriction of an endomorphism f : E →ₗ E to an endomorphism f' : ↥V →ₗ ↥V. To prove that the restriction f' : ↥U →ₛₗ ↥V of a compact operator f : E →ₛₗ F is compact, apply IsCompactOperator.codRestrict to f ∘ U.subtypeL, which is compact by IsCompactOperator.comp_clm.

theorem IsCompactOperator.restrict' {R₂ : Type u_2} [Semiring R₂] {M₂ : Type u_4} [UniformSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] [T0Space M₂] {f : M₂ →ₗ[R₂] M₂} (hf : IsCompactOperator ⇑f) {V : Submodule R₂ M₂} (hV : ∀ v ∈ V, f v ∈ V) [hcomplete : CompleteSpace ↥V] :

IsCompactOperator ⇑(f.restrict hV)

If a compact operator preserves a complete submodule, its restriction to that submodule is compact.

Note that, following mathlib's convention in linear algebra, restrict designates the restriction of an endomorphism f : E →ₗ E to an endomorphism f' : ↥V →ₗ ↥V. To prove that the restriction f' : ↥U →ₛₗ ↥V of a compact operator f : E →ₛₗ F is compact, apply IsCompactOperator.codRestrict to f ∘ U.subtypeL, which is compact by IsCompactOperator.comp_clm.

theorem IsCompactOperator.continuous {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [IsTopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [IsTopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) :

Continuous ⇑f

def ContinuousLinearMap.mkOfIsCompactOperator {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [IsTopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [IsTopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) :

M₁ →SL[σ₁₂] M₂

Upgrade a compact LinearMap to a ContinuousLinearMap.

Equations

Instances For

@[simp]

theorem ContinuousLinearMap.mkOfIsCompactOperator_to_linearMap {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [IsTopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [IsTopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) :

↑(mkOfIsCompactOperator hf) = f

@[simp]

theorem ContinuousLinearMap.coe_mkOfIsCompactOperator {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [IsTopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [IsTopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) :

⇑(mkOfIsCompactOperator hf) = ⇑f

theorem ContinuousLinearMap.mkOfIsCompactOperator_mem_compactOperator {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [IsTopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [IsTopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) :

mkOfIsCompactOperator hf ∈ compactOperator σ₁₂ M₁ M₂

theorem isClosed_setOf_isCompactOperator {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [IsUniformAddGroup M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] :

IsClosed {f : M₁ →SL[σ₁₂] M₂ | IsCompactOperator ⇑f}

The set of compact operators from a normed space to a complete topological vector space is closed.

theorem compactOperator_topologicalClosure {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [IsUniformAddGroup M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] :

(compactOperator σ₁₂ M₁ M₂).topologicalClosure = compactOperator σ₁₂ M₁ M₂

theorem isCompactOperator_of_tendsto {ι : Type u_1} {𝕜₁ : Type u_2} {𝕜₂ : Type u_3} [NontriviallyNormedField 𝕜₁] [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_4} {M₂ : Type u_5} [SeminormedAddCommGroup M₁] [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [IsUniformAddGroup M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] {l : Filter ι} [l.NeBot] {F : ι → M₁ →SL[σ₁₂] M₂} {f : M₁ →SL[σ₁₂] M₂} (hf : Filter.Tendsto F l (nhds f)) (hF : ∀ᶠ (i : ι) in l, IsCompactOperator ⇑(F i)) :

IsCompactOperator ⇑f