# Documentation

Mathlib.Analysis.InnerProductSpace.Symmetric

# Symmetric linear maps in an inner product space #

This file defines and proves basic theorems about symmetric not necessarily bounded operators on an inner product space, i.e linear maps T : E → E such that ∀ x y, ⟪T x, y⟫ = ⟪x, T y⟫.

In comparison to IsSelfAdjoint, this definition works for non-continuous linear maps, and doesn't rely on the definition of the adjoint, which allows it to be stated in non-complete space.

## Main definitions #

• LinearMap.IsSymmetric: a (not necessarily bounded) operator on an inner product space is symmetric, if for all x, y, we have ⟪T x, y⟫ = ⟪x, T y⟫

## Main statements #

• is_symmetric.continuous: if a symmetric operator is defined on a complete space, then it is automatically continuous.

## Tags #

### Symmetric operators #

def LinearMap.IsSymmetric {𝕜 : Type u_1} {E : Type u_2} [] [] (T : E →ₗ[𝕜] E) :

A (not necessarily bounded) operator on an inner product space is symmetric, if for all x, y, we have ⟪T x, y⟫ = ⟪x, T y⟫.

Instances For
theorem LinearMap.isSymmetric_iff_sesqForm {𝕜 : Type u_1} {E : Type u_2} [] [] (T : E →ₗ[𝕜] E) :

An operator T on an inner product space is symmetric if and only if it is LinearMap.IsSelfAdjoint with respect to the sesquilinear form given by the inner product.

theorem LinearMap.IsSymmetric.conj_inner_sym {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →ₗ[𝕜] E} (hT : ) (x : E) (y : E) :
↑() (inner (T x) y) = inner (T y) x
@[simp]
theorem LinearMap.IsSymmetric.apply_clm {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →L[𝕜] E} (hT : ) (x : E) (y : E) :
inner (T x) y = inner x (T y)
theorem LinearMap.isSymmetric_zero {𝕜 : Type u_1} {E : Type u_2} [] [] :
theorem LinearMap.isSymmetric_id {𝕜 : Type u_1} {E : Type u_2} [] [] :
theorem LinearMap.IsSymmetric.add {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →ₗ[𝕜] E} {S : E →ₗ[𝕜] E} (hT : ) (hS : ) :
theorem LinearMap.IsSymmetric.continuous {𝕜 : Type u_1} {E : Type u_2} [] [] [] {T : E →ₗ[𝕜] E} (hT : ) :

The Hellinger--Toeplitz theorem: if a symmetric operator is defined on a complete space, then it is automatically continuous.

@[simp]
theorem LinearMap.IsSymmetric.coe_reApplyInnerSelf_apply {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →L[𝕜] E} (hT : ) (x : E) :
= inner (T x) x

For a symmetric operator T, the function fun x ↦ ⟪T x, x⟫ is real-valued.

theorem LinearMap.IsSymmetric.restrict_invariant {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →ₗ[𝕜] E} (hT : ) {V : } (hV : ∀ (v : E), v VT v V) :

If a symmetric operator preserves a submodule, its restriction to that submodule is symmetric.

theorem LinearMap.IsSymmetric.restrictScalars {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →ₗ[𝕜] E} (hT : ) :
theorem LinearMap.isSymmetric_iff_inner_map_self_real {V : Type u_6} [] (T : V →ₗ[] V) :
∀ (v : V), ↑() (inner (T v) v) = inner (T v) v

A linear operator on a complex inner product space is symmetric precisely when ⟪T v, v⟫_ℂ is real for all v.

theorem LinearMap.IsSymmetric.inner_map_polarization {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →ₗ[𝕜] E} (hT : ) (x : E) (y : E) :
inner (T x) y = (inner (T (x + y)) (x + y) - inner (T (x - y)) (x - y) - IsROrC.I * inner (T (x + IsROrC.I y)) (x + IsROrC.I y) + IsROrC.I * inner (T (x - IsROrC.I y)) (x - IsROrC.I y)) / 4

Polarization identity for symmetric linear maps. See inner_map_polarization for the complex version without the symmetric assumption.

theorem LinearMap.IsSymmetric.inner_map_self_eq_zero {𝕜 : Type u_1} {E : Type u_2} [] [] {T : E →ₗ[𝕜] E} (hT : ) :
(∀ (x : E), inner (T x) x = 0) T = 0

A symmetric linear map T is zero if and only if ⟪T x, x⟫_ℝ = 0 for all x. See inner_map_self_eq_zero for the complex version without the symmetric assumption.