Let $\mathbb{R}^d$ be the usual d-dimensional Hilbert space. I'm learning about projective limits in the category of LCS with continuous linear maps.
How are the following different; and more importantly for my intuition...what are they each equal to?
- $\projlim_{n \in \mathbb{N}} \mathbb{R}$,
- $\projlim_{n \in \mathbb{N}} \mathbb{R}^n$,
- $\oplus_{n \in \mathbb{N}} \mathbb{R}$?
In each case, the chain maps are the (non-expansive) inclusions of $\mathbb{R}^n$ into $\mathbb{R}^m$ when $n\leq m$.
My guess is that the first is $\ell^2$ and the second is some sort of tensor algebra but I'm not sure about its topology...