Theorems · Theorem · operator theory
LinearPMap.adjoint_apply_eq
∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace 𝕜 F] {T : E →ₗ.[𝕜] F}
[inst_5 : CompleteSpace E],
Dense ↑T.domain →
∀ (y : ↥T.adjoint.domain) {x₀ : E}, (∀ (x : ↥T.domain), inner 𝕜 x₀ ↑x = inner 𝕜 (↑y) (↑T x)) → ↑T.adjoint y = x₀- Cited by
- 2 results in Mathlib
- Foundations
- Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- CompleteSpacestatement and proof · cited by 2,532
- Inner.innerstatement and proof · cited by 1,089
- Densestatement and proof · cited by 359
- LinearPMapstatement and proof · cited by 179
- LinearPMap.domainstatement and proof · cited by 167
- LinearPMap.toFun'statement and proof · cited by 124
Cited by2
Results whose statement or proof uses this declaration.
- LinearPMap.IsFormalAdjoint.le_adjointproof · cited by 0
- ContinuousLinearMap.toPMap_adjoint_eq_adjoint_toPMap_of_denseproof · cited by 0