Hilbert Spaces: What Are They?
Note: this was originally a thread on my twitter. If you want to impress people, you can just say a Hilbert space is just a complete infinite dimensional inner product space and leave it at that, but let's talk about what that actually means. When you first learn about vectors, you talk about them as arrows in space; things with a magnitude and a direction. These are elements of \(\mathbb{R}^n\) where n is the number of dimensions of the space you care about. You also talk about the dot product (or inner product) as a way to tell when vectors are orthogonal. (I'm purposely saying "orthogonal" instead of "perpendicular" here, but when you actually think about arrows, it's the same thing.) As my linear algebra students are about to see, \(\mathbb{R}^n\) is far from the only interesting vector space. A classic example is the space of polynomials of dimension less than or equal to \(n\). We know that Taylor series can be used to represent