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Keywords:
inner product space; two dimensional subspace; projection
inner product space; two dimensional subspace; projection
Summary:
The Blaschke--Kakutani result characterizes inner product spaces $E$, among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace $F$ there is a norm 1 linear projection onto $F$. In this paper, we determine which closed neighborhoods $B$ of zero in a real locally convex space $E$ of dimension at least 3 have the property that for every 2 dimensional subspace $F$ there is a continuous linear projection $P$ onto $F$ with $P(B)\subseteq B$.
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References:
[1] Amir D.: Characterizations of Inner Product Spaces. Birkhäuser Verlag, Basel, Boston, Stuttgart, 1986. MR 0897527 | Zbl 0617.46030
[2] Calvert B., Fitzpatrick S.: Nonexpansive projections onto two dimensional subspaces of Banach spaces. Bull. Aust. Math. Soc. 37 (1988), 149-160. MR 0926986 | Zbl 0634.46013
[3] Fitzpatrick S., Calvert B.: Sets invariant under projections onto one dimensional subspaces. Comment. Math. Univ. Carolinae 32 (1991), 227-232. MR 1137783 | Zbl 0756.52002
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