# a counter-example for the existence of the direct limits in $\mathbf{Ban}$

The direct limit of a direct system does not always exists in the category of Banach spaces and bounded linear maps. The statement can be found in many literatures, but I can hardly find a counter-example in any of them. Can someone give a reference that contains a concret counter-example?

• Please cite some of the sources that make this statement. It makes it hard for us to help you with the literature if we don't know what you are reading. Jul 7, 2021 at 21:34
• For example, The Hitchhiker Guide to Categorical Banach Space Theory, P121, below Theorem 4.1. Jul 7, 2021 at 22:04

I don't know a reference but here is a very simple construction of a counterexample. Consider the direct system of inclusions $$\mathbb{R}\to\mathbb{R}^2\to\mathbb{R}^3\to\dots$$ Suppose this system had a direct limit $$X$$ in the category of Banach spaces and bounded linear maps. Note that $$\mathbb{R}^n$$ is the free Banach space on its $$n$$ standard basis vectors: for any Banach space $$Y$$, there is a unique morphism $$\mathbb{R}^n\to Y$$ for any choice of where to send the standard basis vectors. It follows from the universal property of $$X$$ that $$X$$ would be the free Banach space on a countably infinite family of generators $$(x_n)$$, namely the images of the standard basis vectors under all the inclusions $$\mathbb{R}^n\to X$$ into the direct limit.
In particular, this means that for any sequence $$(c_n)$$ in $$\mathbb{R}$$, there is a morphism $$T:X\to\mathbb{R}$$ such that $$T(x_n)=c_n$$ for each $$n$$. But we can choose $$(c_n)$$ such that this morphism could not be bounded: pick $$c_n$$ such that $$c_n>n\|x_n\|$$ for each $$n$$, for instance. This is a contradiction, and thus the direct limit $$X$$ cannot exist.