Hilbert space: treating functions as vectors

Summary

September 06, 2025 at 06:46 Tags Math The tools of linear algebra are extremely useful when working in Euclidean space (e.g. \mathbb{R}^3). Wouldn’t it be great if we could apply these tools to additional mathematical constructs, such as functions and sequences? Hilbert space allows us to do exactly this - apply linear algebra to functions. Intuition - functions as infinite-dimensional vectors There are several ways to view vectors; a standard interpretation is an ordered list of numbers. Let’s take a vector in \mathbb{R}^3 as an example: \[v = \begin{bmatrix} 1.4 \\ 4.2 \\ -2.14 \end{bmatrix}\] This is a list of three numbers, where each number has an index. v[1] is 1.4, v[2] is 4.2 and so on. Another way to think of a vector is a function, in the strict mathematical sense. A vector in \mathbb{R}^3 is a function with the domain {1,2,3} (the indices) and codomain , or: \[v:\{1,2,3\}\to\mathbb{R}\] Now imagine that our vector is N-dimensional: \mathbb{R}^N. Using the function notation we can write v:\{1,2,\cdots ,N\}\to\mathbb{R}. This works for any N, and in fact it also works for an infinite N. Our vector then simply becomes a function from the natural numbers to the reals: v:\mathbb{N}\to\mathbb{R}. But we can take it even further; what if we allow any real number as an index? Our vector is then v:\mathbb{R}\to\mathbb{R}, or we may just change its name to be more familiar: f:\mathbb{R}\to\mathbb{R}. This "vector" is just a function from the reals to the reals. While we can’t write all the elements down explicitly (there’s an infinite number of them, and most of the indices are irrational which don’t even have a finite representation), we can instead come up with a rule that maps an index to the element. For example: f(x)=x^2 is such a rule. For any given index x, it assigns the value x^2. We’re not used to thinking of functions as vectors, but if we carefully extend some definitions, it’s entirely possible! So, functions can be seen as vectors with infinite dimens...