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Sets invariant under projections onto two dimensional subspaces. (English) Zbl 0756.46010
In this interesting paper the authors prove: If \(B\) is a closed neighborhood of 0 in a real locally convex topological vector space \(X\) of dimension \(\geq 3\). Then for all two dimensional subspaces \(M\) there is a continuous linear projection \(P\) of \(X\) onto \(M\) with \(P(B)\subset B\), if and only if \(B\) is the inverse image under a continuous linear map \(T\) of:
(a) the closed unit ball in an inner product \(H\),
(b) the closed unit ball given by a norm on \(R^ 2\), or
(c) a closed neighborhood of 0 in \(R\).
Reviewer: I.Beg (Islamabad)
MSC:
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
46A03 General theory of locally convex spaces
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