Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, let $\lVert\cdot\rVert$ be the \reftext{def:inner-product-norm-2026a}{induced norm}, let $0_{V}$ be the \reftext{lem:vector-space-basic-identities-2026a}{zero vector}, and let $u,v\in V$. Then the following hold.

\textbf{1. (Symmetry)} $u$ and $v$ are \reftext{def:orthogonal-vectors-2026a}{orthogonal} if and only if $v$ and $u$ are orthogonal.

\textbf{2. (Pythagorean identity)} If $u$ and $v$ are orthogonal, then

\textbf{3. (Zero vector)} $0_{V}$ and $v$ are orthogonal.