Theorems · Theorem · functional analysis
Submodule.finrank_orthogonal_span_singleton
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{n : ℕ} [_i : Fact (Module.finrank 𝕜 E = n + 1)] {v : E}, v ≠ 0 → Module.finrank 𝕜 ↥(𝕜 ∙ v)ᗮ = nIn a finite-dimensional inner product space, the dimension of the orthogonal complement of the span of a nonzero vector is one less than the dimension of the space.
- 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.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Moduleproof · cited by 20,661
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Factstatement and proof · cited by 2,726
- FiniteDimensionalproof · cited by 1,854
- Module.finrankstatement and proof · cited by 1,770
- add_commproof · cited by 1,535
- Submodule.spanstatement · cited by 1,504
- DivisionRingproof · cited by 1,062
Cited by2
Results whose statement or proof uses this declaration.
- OrthonormalBasis.fromOrthogonalSpanSingletonproof · cited by 8
- Submodule.mem_span_singleton_of_inner_eq_zero_of_inner_eq_zeroproof · cited by 0