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DIFFEOMORPHISMS BETWEEN SPHERES AND HYPERPLANES IN INFINITE-DIMENSIONAL BANACH
 

Summary: DIFFEOMORPHISMS BETWEEN SPHERES AND
HYPERPLANES IN INFINITE-DIMENSIONAL BANACH
SPACES
DANIEL AZAGRA
Abstract. We prove that for every infinite-dimensional Banach space X
with a Fr´echet differentiable norm, the sphere SX is diffeomorphic to each
closed hyperplane in X. We also prove that every infinite-dimensional Ba-
nach space Y having a (not necessarily equivalent) Cp
norm (with p
N {}) is Cp
diffeomorphic to Y \ {0}.
In 1966 C. Bessaga [1] proved that every infinite-dimensional Hilbert space
H is C
diffeomorphic to its unit sphere. The key to prove this astonishing
result was the construction of a diffeomorphism between H and H \ {0} be-
ing the identity outside a ball, and this construction was possible thanks to
the existence of a C
non-complete norm in H. T. Dobrowolski [5] devel-
oped Bessaga's non-complete norm technique and proved that every infinite-
dimensional Banach space X which is linearly injectable into some c0() is

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

 

Collections: Mathematics