Studia Scientiarium Mathematicarum Hungarica 32. (1996)

1-2. szám - Slezák B.: Implicit function theorem in uniform spaces

Studia Scientiarum Mathematicarum Hungarica 32 (1996), 1-8 IMPLICIT FUNCTION THEOREM IN UNIFORM SPACES D. SLEZÁK In this paper we prove an implicit function theorem for uniform spaces. As a special case an implicit function theorem is also obtained for metriz- able topological groups. Moreover, an implicit function theorem is proved for Banach space case under the same assumptions as Szilágyi’s generalized implicit mapping theorem in [2] with stronger consequences. Definition 1. Let H,X and Y be nonempty sets, f:HxX—>Y a function and y £ im(/). The function g: H —» X is called the implicit function given by the function /, belonging to the value y and passing through the point (to, xo) € H x X ifg(to) = xo and for every t £ H the equality f(t, g(t)) = = y holds. □ The reader can easily control that the following proposition directly fol­lows from the definitions. PROPOSITION 1. Let H, X,Y and f be the same as above, F := (pr//,/), where pr#: H x X —> H is the projection to H. (i) There exists a function g: H —> X such that f{t,g(t)) — y holds for every t£ H if and only if H x {y} ^ F(H x X). (ii) Let T := {/(<,.) | í € H} and Fr be a right inverse of the function F. The equality Fr(t,y) = (t,[f(t, .)]r(y)) uniquely determines the right inverse [/(^) -)]r of the function f(t,.) for every fixed element t of H. Conversely, if for every point t of the set H a right inverse [f(t, .)]r is given, then the function Fr(t,y) — (t,[f(t, ,)]r(y)) is a right inverse of the function F. (iii) The function g: H —> X is an implicit function of f belonging to y £ T if and only if there exists a right inverse Fr of F which satisfies the equality Fr(.,y) = (idu,g). The function g passes through the point (ío^o) G €HxX if and only if [f(to,-)]r(y) = xo. □ Notation. Let (T, r) be a topological space. V 3)t will note that the set V £ r is a neighbourhood of the point t £ T. Definition 2. Let T be a nonempty set, X a topological space, Y a uniform space, By a base of the uniformity of Y. Let /: T x X —> Y be a function and (íoj^o) €T x X be a point. 1991 Mathematics Subject Classification. Primary 54E40; Secondary 46B20. Key words and phrases. Uniform space, inverse function, implicit function, topological group, Banach space. 0081-6906/96/$ 5.00 ©1996 Akadémiai Kiadó, Budapest MAGYAR TJJOOMÁNYOS AKADÉMIA KÖNYVTÁRA