Reproducing Kernel Hilbert Spaces
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...