Can every Banach space be continuously embedded in a reflexive Banach space? As for example for $ L^\infty(\Omega)$ and $L^2(\Omega)$, where $ \Omega\subset R^n$ is a bounded open set.
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The second example you ask about is a Hilbert space, and is therefore a reflexive Banach space. – Keen-ameteur Mar 12 '19 at 10:43
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I think you can modify Tomek's answer here to show that the answer is "no" (reflexive spaces have equivalent strictly convex renormings). – David Mitra Mar 12 '19 at 12:01
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What is "embedding"? So, for example, is the set-theoretic inclusion $L^\infty[0,1] \to L^2[0,1]$ a "continuous embedding"? You do not require the image to be closed? – GEdgar Mar 12 '19 at 12:49
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By the continuous embedding, I mean that : $|y|{L^2(\Omega)}\le C |y|{L^\infty(\Omega)}, \forall y\in L^\infty(\Omega) $ for some constant $C>0,$ without any constraint on the image. – user380020 Mar 12 '19 at 14:38
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I have seen the Tomek's answer. But the strict convexity property is not preserved by equivalence of norm. – user380020 Mar 12 '19 at 14:49
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Sorry for Keen, here I gave an example of such an Embedding ($L^\infty $ in the Hilbert $L^2$) : it is not a part of the question – user380020 Mar 12 '19 at 21:12