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Let $V$ be an infinite dimensional vector space with basis $\{e_i\}_{i \in \mathbb{N}}$ so that every element can be written as a finite linear combination of the $e_i$'s.

Let $V: = V_0 \supseteq V_1 \supseteq V_2 \supseteq V_3 \supseteq \ldots$ be a descending filtration. Then one can define a basis of open sets given by $\{ v + V_k, v \in V, k \in \mathbb{N}\}$ generating a topology on $V$. The equivalence classes of Cauchy sequences in this topology is defined to the completion of $V$, denoted by $\widetilde{V}.$

The filtration can also be used to define a pseudo-metric as follows (as described in this post "Topology on completion") , first set $v(\alpha)=\max \{n\in \mathbb{N}\ |\ \alpha\in V_n\}$, a pseudo-valuation and then $$d(x,y)=2^{-v(x-y)}.$$ It defines the same topology as the one described above.

What is a basis for the completed vector space $\widetilde{V}$? Is there a Hamel or Schauder basis? How can I explicitly describe elements of the vector space $\widetilde{V}$.

winawer
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  • Cauchy sequences make sense in a metric space, not in a general vector space. Is your topology metrisable? In general there's no reason to expect that a metric completion will have an explicit Hamel basis. – Asaf Karagila Aug 04 '21 at 08:04
  • @AsafKaragila: I can use the filtration to define a pseudo-metric that always gives the topology described. I have added more details in my post. Can the completion have a Schauder basis? My naive guess is that the Hamel basis ${e_i}$ is a Schauder basis for the completion, so that elements of the completion are infinite linear combination of the $e_i$s. Does this make sense? – winawer Aug 04 '21 at 18:01

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