Decomposition of mappings approximately inner product preserving

For Hilbert spaces E and F we consider the class of mappings f:E→F preserving approximately inner product in the following sense: |〈f(x)|f(y)〉-〈x|y〉|⩽ϕ(x,y),x,y∈Efor suitable function ϕ. We show that the orthogonal projection of f onto some closed linear subspace H of F is a linear isometry, while the projection onto H⊥ is bounded by ϕ(x,x). Several consequences of such a decomposition are given. In particular, stability and superstability of inner product preserving mappings are considered.