avewright/proofwiki-math huggingface dataset:
Theorem: Densely-Defined Linear Operator with Real-Valued Quadratic Form is Symmetric

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\struct {\map D T, T}$ be a densely-defined linear operator such that:
:$\innerprod {T x} x \in \R$ for each $x \in \map D T$.
Then $T$ is symmetric.

Proof:
Define $q : \map D T \times \map D T \to \C$ by:
:$\map q {x, y} = \innerprod {T x} y$ for each $\tuple {x, y} \in \map D T \times \map D T$.
To show that $T$ is symmetric, we establish that:
:$\map q {x, y} = \overline {\map q {y, x} }$ for each $\tuple {x, y} \in \map D T \times \map D T$.
This will show that:
:$\innerprod {T x} y = \overline {\innerprod {T y} x} = \innerprod x {T y}$ for each $\tuple {x, y} \in \map D T \times \map D T$.
Define $Q : \map D T \to \R$ by:
:$\map Q x = \innerprod {T x} x$ for each $x \in \map D T$.
By hypothesis, we have $\map Q x \in \R$ for each $x \in \map D T$.
By Polarization Identity: Complex Vector Space:
:$4 \map q {x, y} = \map Q {x + y} - \map Q {x - y} + i \map Q {x + i y} - i \map Q {x - i y}$
and:
:$4 \map q {y, x} = \map Q {y + x} - \map Q {y - x} + i \map Q {y + i x} - i \map Q {y - i x}$
for each $\tuple {x, y} \in \map D T \times \map D T$.
We have:
{{begin-eqn}}
{{eqn	| l = \map Q {\lambda x}
		| r = \innerprod {\map T {\lambda x} } {\lambda x}
}}
{{eqn	| r = \innerprod {\lambda T x} {\lambda x}
		| c = $T$ is linear
}}
{{eqn	| r = \cmod \lambda^2 \innerprod {T x} x
		| c = Inner Product is Sesquilinear
}}
{{eqn	| r = \cmod \lambda^2 \map Q x
}}
{{end-eqn}}
Hence we have:
:$\map Q {x - y} = \map Q {y - x}$
We also have:
{{begin-eqn}}
{{eqn	| l = \map Q {x + i y}
		| r = \map Q {i \paren {-i x + y} }
}}
{{eqn	| r = \cmod {i}^2 \map Q {y - i x}
}}
{{eqn	| r = \map Q {y - i x}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn	| l = \map Q {x - i y}
		| r = \map Q {-i \paren {i x + y} }
}}
{{eqn	| r = \cmod {-i}^2 \map Q {i x + y}
}}
{{eqn	| r = \map Q {i x + y}
}}
{{end-eqn}}
Hence we have:
:$\map q {x, y} = \map Q {x + y} - \map Q {x - y} + i \map Q {y - i x} - i \map Q {y + i x}$
Since $\map Q {y - i x}, \map Q {y + i x} \in \R$, we have, by Sum of Complex Conjugates and Product of Complex Conjugates:
:$\map Q {x + y} - \map Q {x - y} + i \map Q {y - i x} - i \map Q {y + i x} = \overline {\map Q {x + y} - \map Q {x - y} + i \map Q {y + i x} - i \map Q {y - i x} } = \overline {\map q {y, x} }$
Hence:
:$\map q {x, y} = \overline {\map q {y, x} }$ for each $\tuple {x, y} \in \map D T \times \map D T$
as required.
{{qed}}
Category:Densely-Defined Linear Operators
Category:Symmetric Densely-Defined Linear Operators
p103k939a519onyag8nz824mkn3kr5p
Theorem: Positive Linear Operator is Symmetric

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\struct {\map D T, T}$ be a positive linear operator.
Then $T$ is symmetric.

Proof:
From the definition of a positive linear operator, we have:
:$\innerprod {T x} x \in \R$ for each $x \in \map D T$.
Hence, from Densely-Defined Linear Operator with Real-Valued Quadratic Form is Symmetric, we have that $T$ is symmetric.
{{qed}}
Category:Symmetric Densely-Defined Linear Operators
b5zozm4pd19qllk82jw5e8phdpfwce2