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## Hilbert spaces (and dagger categories)

Hilbert spaces are a particularly nice class of Banach spaces. They axiomatize ideas from Euclidean geometry such as orthogonality, projection, and the Pythagorean theorem, but the ideas apply to many infinite-dimensional spaces of functions of interest to various branches of mathematics. Hilbert spaces are also fundamental to quantum mechanics, as vectors in Hilbert spaces (up to phase) describe (pure) states of quantum systems.

Today we’ll develop and discuss some of the basic theory of Hilbert spaces. As with the theory of Banach spaces, there are (at least) two types of morphisms we might want to talk about (unitary operators and bounded operators), and we will discuss an elegant formalism that allows us to talk about both. Things written by John Baez will be cited excessively.

## Banach spaces (and Lawvere metrics, and closed categories)

One annoying feature of the abstract theory of vector spaces, and one that often trips up beginners, is that it is not possible to make sense of an infinite sum of vectors in general. If we want to make sense of infinite sums, we should probably define them as limits of finite sums, so rather than work with bare vector spaces we need to work with topological vector spaces over a topological field, usually $\mathbb{R}$ or $\mathbb{C}$ (but sometimes fields like $\mathbb{Q}_p$ are also considered, e.g. in number theory). Common and important examples include spaces of continuous or differentiable functions.

Today we’ll discuss a class of topological vector spaces which is convenient to work with but which still covers many examples of interest, namely Banach spaces. The material in the first half of this post is completely standard and can be found in any text on functional analysis.

In the second half of the post we discuss a category of Banach spaces such that two Banach spaces are isomorphic in this category if and only if they are isometrically isomorphic but which still allows us to talk about bounded linear operators between Banach spaces, and to do this we briefly discuss Lawvere metrics; this material can be found on the nLab.

Let $a, b$ be two $n \times n$ matrices. If $a, b$ don’t commute, then $ab \neq ba$; however, the two share several properties. If either $a$ or $b$ is invertible, then $ab$ is conjugate to $ba$, so in particular they have the same characteristic polynomial.
What if neither $a$ nor $b$ are invertible? As it turns out, $ab$ and $ba$ still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have $ab = 0$ but $ba$ nonzero). There are several ways of proving this result, which implies in particular that $ab$ and $ba$ have the same eigenvalues.
What if $a, b$ are linear transformations on an infinite-dimensional vector space? Do $ab$ and $ba$ still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”