Meredith's Math Blog

 Note: this was originally a thread on my twitter I'm assuming the reader is already familiar with Hilbert spaces. If you need a refresher, here's my post about them. We'll start out in the world of a generic Hilbert space and talk about dual spaces and bounded linear functionals. Conveniently, this is sort of what my linear algebra students are learning about right now! A linear functional is a linear transformation from a vector space to the underlying field: for example, remember \(\ell^2(\mathbb{C})\) :     There are lots of examples of linear functionals, but we'll care about the ones that are *bounded*, that is, the sup of the image of the functional is finite: There's a nice theorem that says that for linear functionals, being bounded is equivalent to being continuous and to being continuous at 0.   The space of continuous linear functionals on a normed vector space is called the dual of the space and has some other nifty properties that I'm not going t...

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 t...