Functions are vectors
I haven&#x27;t read the article yet, but I&#x27;ve known that functions are (infinite) vectors for some years.<p>However, there&#x27;s something that has been bothering me: most of my understanding of linear algebra comes from 2D and 3D spaces, and then in different context of machine learning, datasets that have from tens to even millions of dimensions.<p>In the former, geometric context, the connection between the dimensions are clear: they are orthogonal, but conceptually exactly similar. They are just a 90 degree rotation away from each other.<p>On the other hand, in ML datasets some dimensions are conceptually very similar, and some are totally different. Some are correlated (nearby pixels of an image), some are not, but represent the same unit of quality, and some represent totally different, unrelated things. And as we go toward the mid-layer representations, it becomes very unclear and fuzzy what they represent.<p>In the case of functions, there&#x27;s usually a clear connection between the dimensions: they are of the same unit (the domain and the image (the outputs and inputs) of the function are sets, and those tend to be made of similar-ish – or same type of – things, at least in well-behaved math). And there&#x27;s often a similarity metric between the elements of the sets.<p>The 2D&#x2F;3D linear algebra that I know doesn&#x27;t bother with the &quot;connectedness&quot; of the input dimensions; it only cares the connections from the inputs to the outputs. But surely there is a lot of interesting math that is concerned with the connectedness of the input and output spaces themselves, in context of there still existing a mapping between the input and output. What is that field of math called? What are the interesting results, theorems and such? I love learning more so I&#x27;m kind of just looking for some pointers&#x2F;keywords.