\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}}
\renewcommand{\b}[1]{\boldsymbol{#1}}
\newcommand{\t}[1]{\textsf{#1}}
\newcommand{\m}[1]{\mathbb{#1}}
\def\RR{\bf R}
\def\bold#1{\bf #1}
\def\mbf#1{\mathbf #1}
\def\uv#1{\hat{\usf {#1}}}
\def\dl#1{\underline{\underline{#1}}}
\newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}}
\def\bs#1{\usf #1}\]
A real inner product space
\[\newcommand{\b}[1]{\boldsymbol{#1}}
\newcommand{\pr}[1]{\left(#1\right)}\]
An real inner product space is a real vector space on which the following operation is additionally defined.
\(\langle \cdot, \cdot \rangle : \mathbb{V}\times \mathbb{V} \to \mathbb{R}\).
This operation satisfies the following properties:
For \(\b{u}\), \(\b{v}\), \(\b{z}\in \mathbb{V}\) and for \(a\in \mathbb{R}\), we have that
- \(\langle \b{u}, \b{v} \rangle=\langle \b{v}, \b{u} \rangle\)(symmetry/commutative)
- \(\langle a\b{u}, \b{v} \rangle=\langle \b{u}, a\b{v}\rangle = a \langle\b{u},\b{v}\rangle\) (associative)
- \(\langle \b{u}, \b{v}+\b{z}\rangle =\langle \b{u}, \b{v}\rangle + \langle \b{u}, \b{z}\rangle\) (distributive)
- If \(\langle \b{u}, \b{v} \rangle= 0\) for arbitrary \(\b{u}\) then \(\b{v}=\b{0}\).