# Article

Full entry | PDF   (0.2 MB)
Keywords:
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\$.
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

Partner of