Vector and function

So I’ve been thinking about the great talk by Shaowei (great ideas always make us think). Apart from the sheaf, another important thing for me is the fact that vector and function are the same thing. A function is an infinite-dimension vector, while a vector is a function that maps from {1, 2, …, n} to .

I came to understand that a function is an infinite dimension vector while studying Gaussian process (a Gaussian process is a Multivariate Normal Distribution with infinite dimension). After Shaowei’s talk, I came to understand the other way round. Now, I have the complete picture.

But a question I had is that if vector and function are the same thing, how come one is used to represent a point and the other is used to represent a collection of points? Well, we use these abstract concepts to represent specific things, but the concepts are not the things they represent. Furthermore, I reckon the symbols tell us that one point in an infinite-dimension space is the same as an infinite number of points in a one-dimensional space. I couldn’t see this connection between this two geometric objects when a coordinate system was in my head, but abstracting that out, the symbols showed the way. And with that thought, it sort of makes sense to get rid of the coordinate system….

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