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 metrizable 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. 0081-6906/96/$ 5.00 ©1996 Akadémiai Kiadó, Budapest MAGYAR TJJOOMÁNYOS AKADÉMIA KÖNYVTÁRA 1991 Mathematics Subject Classification. Primary 54E40; Secondary 46B20. Key words and phrases. Uniform space, inverse function, implicit function, topological group, Banach space.