$\ell^1$ functor as left adjoint to unit ball functor

Question

In a comment to this answer

https://mathoverflow.net/a/38755/1106

Yemon Choi notes that "The $\ell^1$ functor is the free Banach space functor, left adjoint to the forgetful unit ball functor".

This statement is intriguing to me, but I am not quite sure what categories and functors Yemon is talking about here. I imagine that we have the category of Banach spaces (with which maps?) on one side, but what about the other? If anyone can fill in the details here I would be very pleased. Also, if anyone has references to a functional analysis text which takes this perspective, I would be happy with that answer as well.

Answer 1

You want to take the category $\text{Ban}_1$ of Banach spaces and short maps (linear maps of operator norm $\le 1$). The unit ball functor $U : \text{Ban}_1 \to \text{Set}$ is represented by $\mathbb{C}$, and its left adjoint sends a set $S$ to the coproduct of $S$ copies of $\mathbb{C}$, which turns out to be $\ell^1(S)$. This says that we have a natural bijection

$$\text{Hom}_{\text{Ban}_1}(\ell^1(S), B) \cong \text{Hom}_{\text{Set}}(S, U(B))$$

which says that a map from a set $S$ to the unit ball $U(B)$ of a Banach space extends uniquely and freely to a short map $\ell^1(S) \to B$, by "linearity."

Intuitively speaking this says that $\ell^1(S)$ is obtained from $S$ by requiring that each element of $S$ have norm $1$ (so that it's in the unit ball and can map shortly to any other element of any other unit ball) and then asking that a linear combination $\sum c_s s$ have the largest possible norm compatible with this (so that it can map shortly to any other such linear combination in any other Banach space). We have $ \| \sum c_s s \| \le \sum |c_s|$ by the triangle inequality and the $\ell^1$ norm is the equality case of this.

This construction generalizes to the construction of the coproduct in $\text{Ban}_1$, which looks like this: if $B_i$ is a collection of Banach spaces, their coproduct in $\text{Ban}_1$ is the completion of the vector space direct sum $\bigoplus_i B_i$ with respect to the "$\ell^1$ norm" $\sum_i \| b_i \|_{B_i}$.

Apologies for the self-promotion, but I go into a bit more detail about categorical properties of $\text{Ban}_1$ (e.g. it is complete, cocomplete, and closed symmetric monoidal) in my blog post Banach spaces (and Lawvere metrics, and closed categories). In particular I attempt to motivate the use of short maps. Note that if we only work with bounded linear maps then we can't hope to recover a Banach space up to isometry via a universal property, whereas the isomorphisms in $\text{Ban}_1$ are isometric. On the other hand the categorical language is still capable of talking about bounded maps, via the closed structure.

Answer 2

Let Bang (Ban, geometric) denote the category whose objects are Banach spaces and whose morphisms are the linear maps that have norm $\leq 1$. (We can work over either real or complex scalars.) Let Set be the category whose objects are sets and whose morphisms are functions.$\newcommand{\Ball}{{\sf ball}}$

There is a functor $\Ball$ from Bang to Set which assigns to each Banach space its closed unit ball; the condition on the morphisms of Bang ensures that each $f:X\to Y$ in Bang restricts to a function $\Ball(X) \to \Ball(Y)$.

What would a left adjoint to $\Ball$ look like? We can use the description/characterization in terms of initial objects in comma categories. So for each set $S$ we want a Banach space $F(S)$ and a function $\eta_S: S \to\Ball(F(S))$ with the following universal property: whenever $E$ is a Banach space and $h:S\to \Ball(E)$ is a function, there is a unique Bang-morphism $T: F(S)\to \Ball(E)$ such that $\Ball(T)\circ\eta_S=f$ as functions.

Unravelling the definitions of the various morphisms: what we require is that for any function $h$ from $S$ to $E$ satisfying $\Vert h(j)\Vert \leq 1$ for all $j\in S$, there should be a unique linear map $T: F(S) \to E$ such that $\Vert T(v)\Vert \leq \Vert v\Vert$ for all $v\in F(S)$ and $T(\eta_S(j))=h(j)$ for all $j\in S$.

Having tried to motivate things, let's make the Ansatz. Define $F(S)$ to be the Banach space $\ell_1(S)$ with its usual norm $\Vert\quad\Vert_1$; let $(e_j)_{j\in S}$ denote the canonical basis bectors in $\ell_1(S)$. The only possible candidate for the linear map $T:\ell_1(S) \to E$ is: define $T(e_j):= h(j)$ for each $j$, and extend by linearity and continuity. To see that this works, observe that for any $v=\sum_{j\in S} \lambda_j e_j \in \ell_1(S)$ we have

$$ \Vert \sum_{j\in S} \lambda_j h(j) \Vert \leq \sum_{j\in S} \vert \lambda_j \vert \Vert h(j)\Vert \leq \sum_{j\in S} \vert \lambda_j \vert \sup_{j\in S} \Vert h(j)\Vert \leq \Vert v \vert_1 $$

Summing up: essentially what the argument above says is that a bounded linear map from $\ell_1(S)$ to a Banach space $E$ defines a bounded function $S\to E$, and that conversely every bounded function $S\to E$ has a unique bounded-linear extension $\ell_1(S)\to E$. (Note that this paragraph, which is stated in analyst language rather than categorist language,is a little more general because I'm not requiring everything to have norm $\leq 1$; but restricting to Bang seems essential if one wants to get a nice statement of this analysis-fact in the language of adjunctions.)

Actually we can go further and say that the adjunction isomorphism $Set(S, \Ball(E)) \cong {\rm Bang}(\ell_1(S),E)$, which a priori is just a naturally-behaving bijection of sets, can be enriched to an isomorphism in Bang: $\ell_\infty(S;E) \cong {\mathcal B}(\ell_1(S),E)$.

Answer 3

This is Exercise 20, on page 167 in Lectures and Exercises on Functional Analysis by Helemskii.

A more ample discussion is carried out by Jiří Rosický in Are Banach spaces monadic?, arXiv:2011.07543.