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

Question

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?

Answer 1

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.