As a preparation for the introduction of the structure of a differentiable
manifold on H\l, Ì) , modelled on a Hubert space, we first consider the
infinitesimal approximation of Ã'°°( Ë Ì) a t c G
M) given by the vector
space of piecewise differentiable vector fields along c. This vector Space can be
viewed as the 'tangent Space'
M) of the 'manifold'
M) at the
point c. Our main interest lies in its completion with respect to an i/'-norm. To
get a precise formulation, we take TcC,0O{L Ì ) to be the space of sections in the
induced bündle c*r.
1.4 DEFINITION. Let c e C/oc(7, M).
(i) Define c*r to be the induced bündle over I:
c*TM - ^ TM
/ - Ì
For each closed subinterval Ij C I with c \ Iy differentiable, the restriction of c*r
to 7y is the (differentiable) induced bündle. The fibre c*r~l(t) over t is also
denoted by TCJ.
(ii) By Cfx(c*TM) we denote the vector space of piecewise differentiable
sections of C*T. We also write TcCf0C(L M) instead and call this the tangent space
toC,00(7, M)atc .
(iii) Using the scalar product (, on the fibres TCJ of C*T, stemming from the
Riemannian metric on the corresponding TiV)M, we define, for £, ç in C'°°(c*TM):
(a) ll€IL = sup|€(')l·
/ e /
(b) ano=/#€(o^(o^
(c) €.ç é = ^ ï + í ß , í ç ï .
The norm derived from the scalar product ( , )
is denoted by || ||
, r = 0,1.
(iv) The completion of Cf0C(c*TM) with respect to the norms || H^ and || ||r is
denoted by C°(c*TM) and H\c*TM\ r = 0,1, respectively.
Á piecewise differentiable section of
is a continuous map ae : / -*
c*TM with c*r ï î = id and such that there exists a subdivision of / into closed
intervals J with c \ J differentiable and ae \ J differentiable. For î \ J we have the
covariant derivative V i | / with respect to the induced connection on
Alternatively, î , or rather T*C Ï £, can be viewed as a piecewise differentiable
vector field along c.
Note that C°(c*TM) is a Banach Space, whereas
r = 0,1, are
Hubert Spaces.
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