English
The map T.reApplyInnerSelf is a continuous linear functional on E; in fact it is continuous when viewed as a linear functional from E to ℝ.
Русский
Функция T.reApplyInnerSelf непрерывна и линейна по отношению к E; является непрерывной функциональностью векторного пространства.
LaTeX
$$Continuous (T.reApplyInnerSelf)$$
Lean4
/-- The adjoint as a linear map from its domain to `E`.
This is an auxiliary definition needed to define the adjoint operator as a `LinearPMap` without
the assumption that `T.domain` is dense. -/
def adjointAux : T.adjointDomain →ₗ[𝕜] E
where
toFun y := (InnerProductSpace.toDual 𝕜 E).symm (adjointDomainMkCLMExtend hT y)
map_add' x
y :=
hT.eq_of_inner_left fun _ => by
simp only [inner_add_left, Submodule.coe_add, InnerProductSpace.toDual_symm_apply, adjointDomainMkCLMExtend_apply]
map_smul' _
_ :=
hT.eq_of_inner_left fun _ => by
simp only [inner_smul_left, Submodule.coe_smul_of_tower, RingHom.id_apply, InnerProductSpace.toDual_symm_apply,
adjointDomainMkCLMExtend_apply]
