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inner product space

Table of Contents

1. Introduction

1. Introduction

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

\begin{aligned}\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{aligned}

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

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