ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4705 on D34: Inner product

Inner product with repeating argument is a real number

Formulation 0

Let $I$ be an D1128: Inner product space over $\mathbb{C}$ such that

(i)	$\langle \cdot, \cdot \rangle$ is the D34: Inner product in $I$
(ii)	$x \in I$ is a D1129: Vector in $I$

Then \begin{equation} \langle x, x \rangle \in \mathbb{R} \end{equation}

Proofs

Proof 0

Let $I$ be an D1128: Inner product space over $\mathbb{C}$ such that

(i)	$\langle \cdot, \cdot \rangle$ is the D34: Inner product in $I$
(ii)	$x \in I$ is a D1129: Vector in $I$

By conjugate symmetry, we have \begin{equation} \langle x, x \rangle = \overline{\langle x, x \rangle} \end{equation} Thus, this result is a consequence of result R2413: Complex conjugate of real number. $\square$