It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any $x\in H$. In an arbitrary norm linear space $X$, if $Y$ is a finite dimensional subspace of $X$ then it also has a closed direct sum complement. I am trying to find two counterexamples

$(1)$ A closed subspace $Y$ in a norm space $X$ (not Hilbert) might not have nearest point for some points $x\notin Y$.

$(2)$ In a norm space $X$, if the assumption finite dimensional of $Y$ is invalid then $Y$ might not admit closed complement.

Thanks in advance!

  • $\begingroup$ Both were asked and answered before: (1) here, (2) here. $\endgroup$ – user127096 Mar 16 '14 at 16:25
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    $\begingroup$ Yes, $(2)$ is Ok. In $(1)$ I want to show that even subspaces (a very special case of convex subsets) also can not admit nearest point. $\endgroup$ – Richkent Mar 16 '14 at 16:40
  • $\begingroup$ Which is what the answers to that question do. $\endgroup$ – user127096 Mar 16 '14 at 16:42
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    $\begingroup$ I don't think that the answers in your first link actually answer the precise question asked in (1). I don't think that any of the examples there are subspaces. $\endgroup$ – Chris Janjigian Mar 16 '14 at 17:30
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    $\begingroup$ @ChrisJanjigian OK, so they are affine subspaces that don't realize the distance to the origin; same picture translated by a vector. $\endgroup$ – user127096 Mar 16 '14 at 19:30

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