1
$\begingroup$

Here is a naïve question: Let $K$ an absorbing, symmetric and convex set of a vector space $X$ that contains 0 that is bounded in the sense that for any direction $x\not=0$, there exists some $n$ such that $nx \not \in K$. Then the Minkowski functional $p_K(x) = \inf \{ \lambda>0: \lambda^{-1} x \in K \}$ defines a norm on $X$.

Is there some criterion that ensures completeness of $X$ with this norm?

Improve this question
$\endgroup$
2
  • $\begingroup$ Such sets $K$ whose Minkowski functional is a complete norm are calle Banach balls (or sometimes Banach discs). A simple but important condition is compactness in some coarser Hausdorff locally convex topology (this is covered by Parschallen's answer). $\endgroup$
    – Jochen Wengenroth
    Commented Jun 22, 2018 at 10:36
  • $\begingroup$ thank you! I just realise, that, with other words, the question was already discussed, for example here :mathoverflow.net/questions/56912/… . Without surprise, the answers resemble :) $\endgroup$
    – Bernhard
    Commented Jun 22, 2018 at 11:05

1 Answer 1

Reset to default
2
$\begingroup$

There is a simple sufficient condition---that there exists a suitable locally convex topology on the space which is weaker than your norm topology and for which $K$ is complete. This is the Grothendieck completeness theorem.

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ What you write is okay (and quite elementary) but I don't think that this is usually called Grothendieck's completeness theorem. There is a theorem of Grothendieck about the completion of a locally convex space (the space of linear functional on the continuous dual with weak$^*$-continuous restrictions to all equi-continuous sets endowed with the topology of uniform convergence on those sets is a completion). $\endgroup$
    – Jochen Wengenroth
    Commented Jun 22, 2018 at 10:33

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .