inner product space

An inner product space is a normed vector space with an inner product defined. This inner product obeys the following properties:

\begin{align} \label{} \langle x,y \rangle = \overline{\langle y,x \rangle} \\ \langle ax + by, z \rangle = a\langle x,z \rangle + b\langle y,z \rangle \\ \langle x,x \rangle > 0, x > 0 \\ \langle x,x \rangle = 0, x = 0 \end{align}

where \(\overline{\langle y,x \rangle}\) is the complex conjugate of \(\langle x,y \rangle\). This gives rise to a normed vector space:

\begin{align} \label{} \lVert x \rVert = \langle x,x \rangle \end{align}