Geodesic
Sets invariant under projections onto two dimensional subspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 233-239 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $P(B)\subseteq B$.
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 $P(B)\subseteq B$.
Classification : 46A03, 46A55, 46C05, 46C15, 52A07, 52A15
Keywords: inner product space; two dimensional subspace; projection 
@article{CMUC_1991_32_2_a4,
     author = {Fitzpatrick, Simon and Calvert, Bruce},
     title = {Sets invariant under projections onto  two dimensional subspaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {233--239},
     year = {1991},
     volume = {32},
     number = {2},
     mrnumber = {1137784},
     zbl = {0756.46010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a4/}
}
TY  - JOUR
AU  - Fitzpatrick, Simon
AU  - Calvert, Bruce
TI  - Sets invariant under projections onto  two dimensional subspaces
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 1991
SP  - 233
EP  - 239
VL  - 32
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a4/
LA  - en
ID  - CMUC_1991_32_2_a4
ER  - 
%0 Journal Article
%A Fitzpatrick, Simon
%A Calvert, Bruce
%T Sets invariant under projections onto  two dimensional subspaces
%J Commentationes Mathematicae Universitatis Carolinae
%D 1991
%P 233-239
%V 32
%N 2
%U http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a4/
%G en
%F CMUC_1991_32_2_a4
Fitzpatrick, Simon; Calvert, Bruce. Sets invariant under projections onto  two dimensional subspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 233-239. http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a4/

[1] Amir D.: Characterizations of Inner Product Spaces. Birkhäuser Verlag, Basel, Boston, Stuttgart, 1986. | MR | Zbl

[2] Calvert B., Fitzpatrick S.: Nonexpansive projections onto two dimensional subspaces of Banach spaces. Bull. Aust. Math. Soc. 37 (1988), 149-160. | MR | Zbl

[3] Fitzpatrick S., Calvert B.: Sets invariant under projections onto one dimensional subspaces. Comment. Math. Univ. Carolinae 32 (1991), 227-232. | MR | Zbl