Theorems · Theorem · functional analysis
ClosedSubmodule.mem_symplComp_iff
∀ {H : Type u_1} [inst : NormedAddCommGroup H] [ipc : InnerProductSpace ℂ H] {x : H} {S : ClosedSubmodule ℝ H},
x ∈ S.symplComp ↔ ∀ y ∈ S, (inner ℂ y x).im = 0- Cited by
- 1 results in Mathlib
- Foundations
- Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites31
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Complexstatement and proof · cited by 5,565
- mul_oneproof · cited by 3,885
- InnerProductSpacestatement and proof · cited by 3,523
- add_zeroproof · cited by 2,707
- MulZeroClass.mul_zeroproof · cited by 2,091
- one_smulproof · cited by 1,374
- Inner.innerstatement and proof · cited by 1,089
- neg_negproof · cited by 960
- Complex.reproof · cited by 882
- Complex.Iproof · cited by 866
Cited by1
Results whose statement or proof uses this declaration.
- ClosedSubmodule.mulI_orthogonal_eq_symplCompproof · cited by 2