Limits of Topological Vector Spaces

Question

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps.

Define the product space $\mathcal Y_N := Y_1 \times \cdots \times Y_N$, and let $\mathcal Y_\infty := \prod_n Y_n$ denote the cartesian product (equipped with the product topology). Let $\pi_N : \mathcal Y_\infty \to \mathcal Y_N$ denote the projection maps.

Let $F_N : X \to \mathcal Y_N$ denote the product function, defined by $F_N(x) := \big( f_1(x), \cdots, f_N(x) \big)$. Does there exist a continuous linear function $F_\infty : X \to \mathcal Y_\infty$ so that $$F_N = \pi_N \circ F_\infty$$ for all $N$?

The answer seems "obviously yes", but there could be a topological issue that I'm missing. Assume that the spaces are infinite-dimensional.

The more general question:

• "Does the category $\operatorname{TopVectSp}$ of topological vector spaces have limits?"

Answer 1

Yes the category of topological vector spaces is complete, and the limits can be created by both forgetful functors to topological spaces and to vector spaces. The proof is trivial.

Answer 2

There is clearly a unique continuous function $F_{\infty}$ satisfying your equations, since it is inherited from $\textbf{Top}$. The only questions that remain are whether this function, and the projection maps, are linear.

But this fact seems obvious, since in the product spaces, the vector space operations are defined pointwise.

Answer 3

The category of topological vector spaces does indeed have limits. See these notes, for example.