Let $X$ be a Hausdorff topological space. It is well-known that every vector bundle on $X$ possesses a continuous inner product, provided that $X$ is paracompact. Is the converse true? -- Namely assume that $X$ is a Hausdorff space such that every vector bundle on $X$ has a continuous inner product, does this imply that $X$ is paracompact?
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asked Nov 17, 2013 at 11:10
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2$\begingroup$ A good starting point could be the Michael selection theorem en.wikipedia.org/wiki/Michael_selection_theorem. Under your assumption, can one define a continuous selection to any lsc multimap on X with values nonempty closed convex sets of a Banach space? $\endgroup$– Pietro MajerCommented Nov 17, 2013 at 16:55
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$\begingroup$ What type of vector bundles we should associate to a lsc multimap on X ? Under my assumption can we prove a weaker result that each lsc multimap on X with valuse in finite dimensional banach space possesses a continuous selection? $\endgroup$– Ali TaghaviCommented Nov 18, 2013 at 18:55
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