# The relationship between LCTVS and projective limit of a projective family of norm spaces.

To begin with, the projective limit of a projective family of norm spaces is a LCTVS (locally convex topological space).

However, on the other hand, I want to find an example of LCTVS such that it cannot be the projective limit of any projective family of norm spaces. I tried to work on the generating seminorms but I got nothing.

To sum up, my questions are:

1. an example(above-mentioned)
2. LCTVS under what extra conditions could be isomorphic to a projective limit of a projective family of norm spaces?

Mathematics Asked on November 12, 2021

Let $$E$$ be a Hausdorff locally convex space and $${p_alpha: alpha in I}$$ be a directed family of seminorms that defines the topology of E (f.e. it can be the set of all continuous seminorms on $$E$$). Let $$E_alpha = E/p_alpha^{-1}({0})$$ for all $$alpha in I$$. $$E_alpha$$ is a normed space wrt to norm that is induced by $$p_alpha$$ (i.e. $$||pi_alpha(x)||_alpha = p_alpha(x)$$, where $$pi_alpha:E rightarrow E_alpha$$ is the canonical projection and $$x in E$$). If $$p_alpha ge c p_beta$$, where $$c > 0$$, then there is a canonical map $$E_alpha rightarrow E_beta$$ that is continuous wrt to the foregoing norms. Thus, spaces $$E_alpha$$ form a projective family of normed spaces. It can be easily shown that $$E$$ is canonically isomorphic to the projective limit of this projective family (note that if $$E$$ is not necessarily Hausdorff, then this projective limit is canonically isomorphic to $$E/overline{{0}}$$).

Thus, every Hausdorff locally convex space is isomorphic to a projective limit of normed spaces. Since the projective limit of a family of Hausdorff locally convex spaces is again Hausdorff, it follows that the Hausdorff condition is also necessary. (For non-Hausdorff spaces you can define $$E_alpha$$ as a seminormed space that is algebraically equal to $$E$$ with seminorm $$p_alpha$$. Thus, arbitrary locally convex space is a projective limit of a family of seminormed spaces.)

There is a notable addition to the foregoing construction. We can also consider the completions of $$E_alpha$$ and the continuations of the canonical maps $$E_alpha rightarrow E_beta$$. The projective limit of this directed family is the completion of $$E$$. Thus, a Hausdorff locally convex space $$E$$ is a projective limit of a projective family of Banach spaces iff $$E$$ is complete (since projective limit preserves completeness).

Answered by Matsmir on November 12, 2021

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