<story><title>Functions are vectors</title><url>https://thenumb.at/Functions-are-Vectors/</url></story><parent_chain></parent_chain><comment><author>ubj</author><text>I wish I could upvote this twice. This is the best basic introduction to concepts in functional analysis that I&amp;#x27;ve seen. Another great overview that goes deeper into the math is [1].&lt;p&gt;Another fantastic application that the website doesn&amp;#x27;t mention is the composition &amp;#x2F; Koopman operator. In control theory (e.g. autonomous drones, cars, robot arms, etc.), most real-world systems are described by nonlinear dynamics which are very difficult to work with (e.g. safety&amp;#x2F;stability guarantees, optimizing over forward horizons using NMPC, state estimation, etc.) The Koopman operator however gives a globally relevant linear approximation of non-linear systems. In other words, you can treat a nonlinear system as a linear system with fairly high accuracy. This greatly simplifies control and estimation from a computational perspective. You can also learn these linearizations from data. Steve Brunton has some good materials on Koopman theory [2][3], and there are some great applications to control of systems such as soft robots [4].&lt;p&gt;[1]: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;arxiv.org&amp;#x2F;abs&amp;#x2F;1904.02539&quot; rel=&quot;nofollow noreferrer&quot;&gt;https:&amp;#x2F;&amp;#x2F;arxiv.org&amp;#x2F;abs&amp;#x2F;1904.02539&lt;/a&gt;&lt;p&gt;[2]: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;youtube.com&amp;#x2F;playlist?list=PLMrJAkhIeNNSVXUvppZTYNHKQUD-oWys9&quot;&gt;https:&amp;#x2F;&amp;#x2F;youtube.com&amp;#x2F;playlist?list=PLMrJAkhIeNNSVXUvppZTYNHKQ...&lt;/a&gt;&lt;p&gt;[3]: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;arxiv.org&amp;#x2F;abs&amp;#x2F;2102.12086&quot; rel=&quot;nofollow noreferrer&quot;&gt;https:&amp;#x2F;&amp;#x2F;arxiv.org&amp;#x2F;abs&amp;#x2F;2102.12086&lt;/a&gt;&lt;p&gt;[4]: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;arxiv.org&amp;#x2F;abs&amp;#x2F;1902.02827&quot; rel=&quot;nofollow noreferrer&quot;&gt;https:&amp;#x2F;&amp;#x2F;arxiv.org&amp;#x2F;abs&amp;#x2F;1902.02827&lt;/a&gt;</text></comment>
<story><title>Functions are vectors</title><url>https://thenumb.at/Functions-are-Vectors/</url></story><parent_chain></parent_chain><comment><author>jesuslop</author><text>I always liked this viewpoint a lot. I&amp;#x27;m enjoying with abandon some dusty lectures that Vito Volterra gave in Madrid on differential and integrodifferential equations, while helping also to create Functional Analysis (a Functional being the analogue of a dual vector). He is constantly exploiting this analogy method from finite variable constructions to infinite, also uncountable variables. Even up to showing some embarrassment of being too repetitive with the idea! People in teaching should join and take a peek.&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;searchworks.stanford.edu&amp;#x2F;view&amp;#x2F;526111&quot; rel=&quot;nofollow noreferrer&quot;&gt;https:&amp;#x2F;&amp;#x2F;searchworks.stanford.edu&amp;#x2F;view&amp;#x2F;526111&lt;/a&gt;</text></comment>