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diff --git a/mindmap/inner product space.org b/mindmap/inner product space.org new file mode 100644 index 0000000..e9448a0 --- /dev/null +++ b/mindmap/inner product space.org @@ -0,0 +1,26 @@ +:PROPERTIES: +:ID: 36a2715c-a8db-4b75-b799-61ce43be2d2d +:END: +#+title: inner product space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* 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} + |