Theorems · Theorem · functional analysis
sum_inner
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{ι : Type u_4} (s : Finset ι) (f : ι → E) (x : E), inner 𝕜 (∑ i ∈ s, f i) x = ∑ i ∈ s, inner 𝕜 (f i) xAn inner product with a sum on the left.
- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Finsetstatement and proof · cited by 13,712
- Finset.sumstatement · cited by 5,195
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Inner.innerstatement · cited by 1,089
- map_sumproof · cited by 455
- LinearMap.flipproof · cited by 193
- innerₛₗproof · cited by 28
Cited by16
Results whose statement or proof uses this declaration.
- Matrix.toLin_conjTransposeproof · cited by 4
- Finsupp.sum_innerproof · cited by 3
- Matrix.star_dotProduct_gram_mulVecproof · cited by 3
- ProbabilityTheory.charFun_stdGaussianproof · cited by 2
- Orthonormal.sum_inner_products_leproof · cited by 2
- LinearMap.isSymmetric_sumproof · cited by 1
- inner_sum_smul_sum_smul_of_sum_eq_zeroproof · cited by 1
- Orthonormal.inner_left_sumproof · cited by 1
- OrthogonalFamily.inner_sumproof · cited by 1
- LinearMap.isPositive_sumproof · cited by 1
- ProbabilityTheory.covarianceBilin_apply_piproof · cited by 1
- DFinsupp.sum_innerproof · cited by 0