Loogle!

Result

Found 6 declarations mentioning OrthonormalBasis.map.

• OrthonormalBasis.map 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {G : Type u_7} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) : OrthonormalBasis ι 𝕜 G
• Complex.map_isometryOfOrthonormal 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {F : Type u_5} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {F' : Type u_6} [NormedAddCommGroup F'] [InnerProductSpace ℝ F'] (v : OrthonormalBasis (Fin 2) ℝ F) (f : F ≃ₗᵢ[ℝ] F') : Complex.isometryOfOrthonormal (v.map f) = (Complex.isometryOfOrthonormal v).trans f
• OrthonormalBasis.toBasis_map 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {G : Type u_7} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) : (b.map L).toBasis = b.toBasis.map L.toLinearEquiv
• OrthonormalBasis.map_apply 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {G : Type u_7} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) (i : ι) : (b.map L) i = L (b i)
• OrthonormalBasis.map.eq_1 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {G : Type u_7} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) : b.map L = { repr := L.symm.trans b.repr }
• OrthonormalBasis.span.eq_1 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {ι' : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [DecidableEq E] {v' : ι' → E} (h : Orthonormal 𝕜 v') (s : Finset ι') : OrthonormalBasis.span h s = (OrthonormalBasis.mk ⋯ ⋯).map (LinearIsometryEquiv.ofEq (Submodule.span 𝕜 ↑(Finset.image v' s)) (Submodule.span 𝕜 (Set.range (v' ∘ Subtype.val))) ⋯).symm

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision bce1d65