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## String diagrams, duality, and trace

Previously we introduced string diagrams and saw that they were a convenient way to talk about tensor products, partial compositions of multilinear maps, and symmetries. But string diagrams really prove their use when augmented to talk about duality, which will be described topologically by bending input and output wires. In particular, we will be able to see topologically the sense in which the following four pieces of information are equivalent:

• A linear map $U \to V$,
• A linear map $U \otimes V^{\ast} \to 1$,
• A linear map $V^{\ast} \to U^{\ast}$,
• A linear map $1 \to U^{\ast} \otimes V^{\ast}$.

Using string diagrams we will also give a diagrammatic definition of the trace $\text{tr}(f)$ of an endomorphism $f : V \to V$ of a finite-dimensional vector space, as well as a diagrammatic proof of some of its basic properties.

Below all vector spaces are finite-dimensional and the composition convention from the previous post is still in effect.

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## Introduction to string diagrams

Today I would like to introduce a diagrammatic notation for dealing with tensor products and multilinear map. The basic idea for this notation appears to be due to Penrose. It has the advantage of both being widely applicable and easier and more intuitive to work with; roughly speaking, computations are performed by topological manipulations on diagrams, revealing the natural notation to use here is 2-dimensional (living in a plane) rather than 1-dimensional (living on a line).

For the sake of accessibility we will restrict our attention to vector spaces. There are category-theoretic things happening in this post but we will not point them out explicitly. We assume familiarity with the notion of tensor product of vector spaces but not much else.

Below the composition of a map $f : a \to b$ with a map $g : b \to c$ will be denoted $f \circ g : a \to c$ (rather than the more typical $g \circ f$). This will make it easier to translate between diagrams and non-diagrams. All diagrams were drawn in Paper.

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## 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.

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