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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

1 Answers

One Answer

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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