In this paper we give an estimate for the size of the subset of X consisting of points at which the average of / is equal to zero. The result represents an extension of the Borsuk-Ulam-Yang theorem. Let F be one of the fields R, C or H, and G C F the unit sphere equipped with the standard group structure. Take any space A with a free action of G, let V be a finite-dimensional vector space over F with the standard action of G (multiplication by units), and let /: A-► V be a continuous map. The size of the set of points in X at which the average of / is zero can be described in terms of an invariant called the index [5]. We would like to show that for a certain class of maps, this can be generalized to the case where the representation space V is replaced by an arbitrary Banach space E over F. This has already been done by Spannier and Holm [4] for the field R, that is, if A is a space with a free action of G = Z2. The average of a map / : X-► E, from a space A with a free action of G to a representation space F for G (possibly infinite-dimensional), is the map Av/: X-► E, defined by Av f(x)-fG g"1 f(gx) dg, where / denotes the Haar integral on G. The point ¡r € A is a balanced point of / if Av f(x) = 0. The set of balanced points is denoted by A(f). Proposition l. Let f: X-* F be continuous. (a) Av/ is an equivariant continuous map. (b) /// is equivariant, then Av/ = /. (c) For any map f, A(f) is a closed invariant subset of X and A(f) = A(Av/). PROOF, (a) Av/ is equivariant, since for any h EG, Avf(hx) = f g-1f(ghx)dg = h f (gh)'1 f(ghx)d(gh) = hkvf(x). JG JG The map F: G x X-> E, defined by F(g,x) = f(gx), is continuous. Thus, for any £ > 0 and for any pair (g,x) G G x A, there exist open neighborhoods U9tX of g and Vg,x of x such that \\f(gx)-f(hy)\\ < £ for any h G Ug,x and y e Vg,x.