Let H be an infinite dimensional Hilbert space with a filtration \((H_ n)_{n\in {\mathbb{N}}}\) by finite dimensional subspaces with dense union, let S be the unit sphere in H and call \(S_ n=S\cap H_ n\). A uniformly continuous map \(f: S\to S\) is called a DJ-mapping if \(\lim_{n\to \infty}\sup_{x\in S_ n}d(fx,S_ n)=0\). Here, d is the metric of H. Call [S,S] the set of DJ-homotopy classes of DJ-mappings \(S\to S\). The author proves that there is a (noncanonical) homomorphism of [S,S] into the group \(\prod_{n\in {\mathbb{N}}}G_ n/\oplus_{n\in {\mathbb{N}}}G_ n\) where \(G_ n={\mathbb{Z}}\) if dim \(H_ n-\dim H_{n-1}>1\) and \(G_ n={\mathbb{Z}}\oplus {\mathbb{Z}}\) if dim \(H_ n-\dim H_{n-1}=1\). It is unknown whether the homomorphism is injective. This homomorphism is, in a sense, a generalization of Leray-Schauder degree.