Proof of Symmetry of Orthogonality and the Pythagorean Identity

Conditions 1-4 below are those of Complex Inner Product Space, and we use the elementary identities of Elementary Properties of a Complex Inner Product and the properties of conjugation in Properties of Complex Conjugation and Modulus.

Claim 1. By condition 1, $\langle v,u\rangle=\overline{\langle u,v\rangle}$. If $\langle u,v\rangle=0$, then $\langle v,u\rangle=\overline{0}=0$, since $0$ has real part $0$ and imaginary part $0$ and hence equals its own conjugate. Conversely, if $\langle v,u\rangle=0$, then $\langle u,v\rangle=\overline{\langle v,u\rangle}=\overline{0}=0$ by the same argument with the roles exchanged. So the two conditions are equivalent, which is the assertion about orthogonality.

Claim 2. By additivity in each argument (condition 2 and claim 1 of Elementary Properties of a Complex Inner Product) and the definition of the induced norm,

If $u$ and $v$ are orthogonal, then $\langle u,v\rangle=0$ and, by claim 1, $\langle v,u\rangle=0$, so the middle two terms vanish and

Claim 3. By claim 3 of Elementary Properties of a Complex Inner Product, $\langle 0_{V},v\rangle=0$, so $0_{V}$ and $v$ are orthogonal.