 
Summary: DIFFEOMORPHISMS BETWEEN SPHERES AND
HYPERPLANES IN INFINITEDIMENSIONAL BANACH
SPACES
DANIEL AZAGRA
Abstract. We prove that for every infinitedimensional 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 infinitedimensional 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 infinitedimensional 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
noncomplete norm in H. T. Dobrowolski [5] devel
oped Bessaga's noncomplete norm technique and proved that every infinite
dimensional Banach space X which is linearly injectable into some c0() is
