Consider each Euclidean space $\mathbb{R}^n$ as a topological vector space. I wondered what I get by taking $n \to \infty$ in the category of topological vector spaces over $\mathbb{R}$.
There are really two kinds of limit here; the direct limit and the inverse limit. By letting the inclusion maps $f_n : \mathbb{R}^n \to \mathbb{R}^{n+1}$ be defined as $f_n(x_1, \cdots, x_n) = (x_1, \cdots, x_n, 0)$, I concluded that the direct limit is the real vector space with countably infinite dimension, denoted by $\mathbb{R}^\infty$, endowed with the box topology. By letting the projection maps $g_n : \mathbb{R}^{n+1} \to \mathbb{R}^n$ be defined as $g_n(x_1, \cdots, x_n, x_{n+1}) = (x_1, \cdots, x_n)$, I concluded that the inverse limit is $\mathbb{R}^\omega$ endowed with the product topology.
Does this mean $\mathbb{R}^\infty$ and $\mathbb{R}^\omega$ are Hilbert spaces? That is, are there inner products that induces the topologies?