English
For a LinearIsometryEquiv f between E and E', the starProjection on the mapped subspace corresponds to the starProjection on the original via f and f^{-1}.
Русский
Для линейной изометрной эквивариантности f между E и E' проекция на отображение сохраняется через f и f^{-1}.
LaTeX
$$P_{(p.map f.toLinearMap)}^{*}(x) = f(P_p^{*}(f^{-1}(x))).$$
Lean4
@[simp]
theorem range_starProjection (U : Submodule 𝕜 E) [U.HasOrthogonalProjection] : LinearMap.range U.starProjection = U :=
by
ext x
exact ⟨fun ⟨y, hy⟩ ↦ hy ▸ coe_mem (U.orthogonalProjection y), fun h ↦ ⟨x, starProjection_eq_self_iff.mpr h⟩⟩