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:ID:       36a2715c-a8db-4b75-b799-61ce43be2d2d
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#+title: inner product space
#+author: Preston Pan
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* Introduction
An inner product space is a [[id:9a1cc2d9-ef99-436c-8c21-9e68fd7df192][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}