Real and complex inner-product spaces and their induced length

Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$, with conjugation $z\mapsto \bar{z}$ the identity on $\mathbb{R}$ and complex conjugation on $\mathbb{C}$ (Real and imaginary parts, complex conjugation, and modulus). A real or complex inner-product space is an $\mathbb{F}$-vector space $V$ together with an inner product $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$ in the sense of Real and complex inner product spaces, with the inner product linear in the first argument: for all $u,v,w\in V$ and $a,b\in \mathbb{F}$,

$⟨au+bv,w⟩=a⟨u,w⟩+b⟨v,w⟩,⟨u,v⟩=\overline{⟨v,u⟩},⟨v,v⟩\ge 0,⟨v,v⟩=0⟺v=0.$

Conjugate-linearity in the second argument. Conjugate symmetry together with linearity in the first argument gives, for all $w\in V$,

$⟨w,au+bv⟩=\overline{⟨au+bv,w⟩}=\overline{a⟨u,w⟩+b⟨v,w⟩}=\bar{a}⟨w,u⟩+\bar{b}⟨w,v⟩\mathrm{.}$

Induced length. The induced length, or inner-product norm, of $v$ is

$\mathrm{\parallel }v\mathrm{\parallel }:=\sqrt{⟨v,v⟩},$

which is exactly the published The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: positive definiteness makes $⟨v,v⟩$ a nonnegative real with a unique nonnegative square root, so that $\mathrm{\parallel }v\mathrm{\parallel }\ge 0$, $\mathrm{\parallel }v\mathrm{\parallel }=0$ exactly for $v=0$, and $⟨v,v⟩=\mathrm{\parallel }v{\mathrm{\parallel }}^2$.

The convention on this page. Every inner-product space below is real or complex in this sense, the pairing is linear in the first argument and conjugate-linear in the second, and $\mathrm{\parallel }v\mathrm{\parallel }$ always denotes the induced length. In the real case conjugation is the identity and $\mathrm{\mid }z\mathrm{\mid }$ is the absolute value of $\mathbb{R}$; in the complex case they are complex conjugation and the complex modulus. All statements below are therefore written once and read in either scalar field.

Real and imaginary parts, complex conjugation, and modulus

Real and complex inner product spaces, with the inner product linear in the first argument

The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product