$\ell^2$ as colimit in $\mathbf{TopVect}_{\mathbb{R}}$

Question

Let $\mathbf{TopVect}_{\mathbb{R}}$ be the category of topological vector spaces with continuous linear maps as morphisms. Is it ineed true that $\ell^2 \cong \varinjlim_{n}\oplus_{i=1}^n\mathbb{R}$?

Answer 1

While Zvi's given a good answer for the exact question the OP asked. I suspect the OP might be interested in whether or not it is possible to view $\ell^2$ as a colimit.

The problem with trying this with $\mathbf{TopVect}_\Bbb{R}$ is that it's too big. In order to end up with an uncountably infinite dimensional vector space, we need a restriction on the spaces allowed in the category that forces the space to be e.g. complete.

Hence, as one example of a category making $\ell^2$ the colimit, we can take the category of real Hilbert spaces with unitary maps as morphisms.

Then if we have unitary maps $T_n : \Bbb{R}^n \to H$ for some Hilbert space $H$, we can define $T:\ell^2 \to H$ by $T(e_i) = T_i(e_i)$ (and this is the only possible valid definition of $T$). It's fairly obvious that this is well defined, since all of the maps $T_i$ are unitary, so we'll end up with $\{T_i(e_i)\}$ being a countably infinite orthonormal set in $H$.

It might be possible to show that $\ell^2$ is the colimit in a slightly larger appropriate category, like Banach spaces with norm preserving maps, but I think this is enough already.

Answer 2

Any direct limit of $\Bbb R^n$ is a countable dimensional vector space over $\Bbb R$. But any $\ell^p_{\Bbb R}$ space for any $p\ge 1$ is uncountable dimensional, so it cannot be a direct limit of $\Bbb R^n$.