The name of this site is a reference to a type of mathematical notation used in quantum mechanics, invented by Paul Dirac. Quantum states can be represented by vectors, so called “ket” vectors and their dual, “bra” vectors (and hence this notation is called Bra-ket notation). Representing quantum states this way makes the math of quantum mechanics much more user-friendly.
The term “ket space” is somewhat non-standard, but is used in a popular quantum mechanics text by Sakurai. More commonly quantum physicists will use the term Hilbert space which is a special type of vector space that has the properties of completeness, an inner product, and can be infinite-dimensional. Simply put, a vector space is a collection mathematical objects with some shared features.
As for why this is the name of the site, I studied quantum mechanics in college and my research in grad school involved using the mathematical apparatus of linear algebra for quantum physics calculations. However there’s really no special reason for my choice of this particular name for the site aside from being a quirky reference to something I’m interested in. I just thought it was a neat domain name and unfortunately Hilbert.space was already taken 🙁