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Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that $k$ is maximally complete. On page 43 of 'Seminaire Banach' published as Springer Lecture notes in Mathematics volume 277, it is assumed that the valuation is non-discrete. What is the main reason behind these restrictions? I am interested in bornological vector spaces over a field with trivial valuation. Banach spaces over such a field make sense and therefore one gets a metric space and therefore a bornological set where the bornology is compatible with the linear structure. This should still be a (complete) bornological vector space hopefully. For what parts of the theory are these extra restrictions needed or useful? Are there some pathologies about the category of bornolocgical vector spaces over a general complete non-archimedean field that are not present when you add these extra assumptions?

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that $k$ is maximally complete. On page 43 of 'Seminaire Banach' published as Springer Lecture notes in Mathematics volume 277, it is assumed that the valuation is non-discrete. What is the main reason behind these restrictions? I am interested in bornological vector spaces over a field with trivial valuation. Banach spaces over such a field make sense and therefore one gets a metric space and therefore a bornological set where the bornology is compatible with the linear structure. This should still be a (complete) bornological vector space hopefully. For what parts of the theory are these extra restrictions needed or useful? Are there some pathologies about the category of bornological vector spaces over a general complete non-archimedean field that are not present when you add these extra assumptions?

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that $k$ is maximally complete. On page 43 of 'Seminaire Banach' published as Springer Lecture notes Mathematics volume 277, it is assumed that the valuation is non-discrete. What is the main reason behind these restrictions? I am interested in bornological vector spaces over a field with trivial valuation. Banach spaces over such a field make sense and therefore one gets a metric space and therefore a bornological set where the bornology is compatible with the linear structure. This should still be a (complete) bornological vector space hopefully. For what parts of the theory are these extra restrictions needed or useful? Are there some pathologies about the category of bornolocgical vector spaces over a general complete non-archimedean field that are not present when you add these extra assumptions?

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that $k$ is maximally complete. On page 43 of 'Seminaire Banach' published as Springer Lecture notes in Mathematics volume 277, it is assumed that the valuation is non-discrete. What is the main reason behind these restrictions? I am interested in bornological vector spaces over a field with trivial valuation. Banach spaces over such a field make sense and therefore one gets a metric space and therefore a bornological set where the bornology is compatible with the linear structure. This should still be a (complete) bornological vector space hopefully. For what parts of the theory are these extra restrictions needed or useful? Are there some pathologies about the category of bornolocgical vector spaces over a general complete non-archimedean field that are not present when you add these extra assumptions?