Decomposition of mappings approximately inner product preserving Original Research Article

For Hilbert spaces E and F we consider the class of mappings f:E→Ff:E→F preserving approximately inner product in the following sense: View the MathML source|〈f(x)|f(y)〉-〈x|y〉|⩽ϕ(x,y),x,y∈E Turn MathJax on for suitable function ϕϕ. 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⊥H⊥ is bounded by View the MathML sourceϕ(x,x). Several consequences of such a decomposition are given. In particular, stability and superstability of inner product preserving mappings are considered.