LECTURE 2. ANALYTICAL TOOLS 65for alls?: 0, EE R Moreover,
z(s, t) = ef;, r(T)dT[e(t) + r(s, t)]
where z = (x, y) and e =/=- 0 is an eigenvector of some selfadjoint operator Ac)O
corresponding to a negative eigenvalue >. < 0 and I: JR+ ----> JR is a smooth function
satisfying
1(s)----> >. ass----> oo.
In particular e(t) =/=- 0 'Vt E JR and the remainder term r satisfies
8°'r(s,t)----> 0for all derivatives a:= (a: 1 , 0::2) uniformly int ER
This theorem is the most important ingredient to a variety of results. For
example Fredholm theory. Another important application are the constructions of
invariants, which are connected with the question if the M-part u is an embedding
or an immersion, see [52].