Embedding CW-complexes into infinite-dimensional topological vector spaces Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s Algebraic Topology for example. But since I learned about this result I was wondering if there are similar results for ‘larger’ vector spaces.
It is well known result that a locally-finite finite-dimensional CW-complex $(X,\mathcal{E},\varphi)$ such that $\lvert\mathcal{E}\rvert\le \aleph_0$ can be homeomorphically embedded into space $\mathbb{R}^{1+2n}$, where $n$ is the dimension of the complex. See for the detailed proof CW-complexes and Euclidean spaces by Rudolf Fritsch and Renzo Piccinini. It seems that in the case of infinite-dimensional complexes it still must be possible to embed it into space $\mathbb{R}^{\oplus \mathbb{N}}$ by considering a direct limit of embeddings of skeleta $X_n \hookrightarrow \mathbb{R}^{k(n)}$  with the final topology (but I may be wrong here).
Moreover, it seems that local-finiteness is required for ensuring local compactness of the embedding. So,  the embeddings of skeleta $X_n \hookrightarrow V_n$ may still exist if $V_n$ are infinite-dimensional topological vector spaces, even if $X$ is not locally finite. Thus, $X$ must have embedding into a direct limit $\varinjlim V_n$, which is hopefully an LF-space. the It also may be the case, but I'm not so sure, that the condition $\lvert\mathcal{E}\rvert\le \aleph_0$ can also be relaxed if one considers spaces which are not second-countable. So, there still may exist CW-complexes which are embeddable into such vector space, but are not embeddable into any Euclidean space for the obvious reasons.
So, the idea is to embed CW-complexes, which are not normally embeddable into euclidean spaces into some ‘very large’ space, which still has vector space structure. So my question:
Is there any obvious reasons why this approach is deemed to fail? Is there any known results on embedding CW-complexes into infinite dimensional spaces? Is there any simple conditions which may prevent a CW-complex from being embeddable into any vector space? (For example not being $T_3$-regular, but I don't know such examples.) Are there any conditions for embeddability of a CW-complex into an arbitrary vector space, but not yet a Euclidean space?
The most dull result possible here is something like: Embedding in topological vector space exists iff embedding in Euclidean space exists. But this result seems to be too strong and strict.
I will be grateful for any pieces of information.
 A: This is a comment but it will be too long—hence the use of answer format.  There is a standard procedure to embed the objects of suitable manifold type categories into complete locally convex spaces in a canonical way.  The basic idea is to consider the free vector space over the object and provide it with the finest l.c. structure such that the natural embedding of the original object therein has a suitable structural property.  One then  completes this l.c.s.  This rather vague description will become clearer if we consider what is perhaps the simplest case—that of the category of topological spaces, more precisely completely regular ones (otherwise there is no hope of embedding in a l.c.s.).   We give the free vector space $\Lambda(X)$ over such a space the finest l.c.s. topology such that the corresponding embedding of $X$ into $\Lambda(X)$ is continuous and then take the completion. There are many examples which can be dealt with similarly—most notably categories of uniform spaces,  smooth manifolds, complex manifolds, piecewise linear manifolds.  The essential element is that there is a clear notion of what it means for a function on an object of the category with values in  a l.c.s. to have the correct structural property (in the above cases, continuity, uniform continuity, smoothness, partial linearity).  These l.c.s.’s then have a suitable universal property which characterises them uniquely and in the above examples, they have natural interpretations as spaces of measures or distributions.
I have never speculated on whether a suitable version of this construction would work in your situatiion—hence the tentative nature of this contribution as a comment rather than answer—but it might be worth a try.
