**Approximation of invariant subsets by invariant subspaces with application to Levi-Civita functional equations**

Let T be a continuous representation of an amenable group G on a linear topological space X. Let Y ⊂ X be a closed subspace of X invariant for T and supplied with a norm ∥·∥_{Y} such that the restriction of T to Y is an isometric representation.

For ξ ∈ X and any subspace X_{0}⊂X, we define the Y-distance from ξ to X_{0} by

d_{Y}(ξ,X_{0}) = inf{∥y∥: y ∈ Y, ξ -y ∈ X_{0}}.

**Statement**: For each ϵ>0 and n∈N there is a δ>0 with the following property:

if L is a finite-dimensional subspace of X with dim(L) ≤ n and

d_{Y}(T_{g}ξ,L) < δ for all g ∈ G

then there is a T-invariant subspace M⊂X with dim(M) ≤ 3n and

d_{Y}(ξ,M) < ϵ.

We apply the result to the study of stability of the Levi-Civita functional equation

f(gh)=∑ _{j=1}^{N} u_{j}(g)v_{j}(h), g, h ∈ G

in the class of measurable functions on an amenable group G.

Stability in the Ulam-Hyers sense means that each function that “almost” satisfies the equation is “close” to a proper solution.

For bounded functions the study of stability of ([1]) leads to the following geometric problem:

given a representation g ↦ T_{g} of a group G on a Banach space X, for any invariant subset K⊂ X to estimate its distance to invariant subspaces of X via the distances to arbitrary n-dimensional subspaces.

In this setting the problem was studied in [1] using a specially developed techniques of covariant width, and the estimate is more strict: dim(M) ≤ dim(L).

- E. Shulman,: Group representations and stability of functional equations,
*J. London Math. Soc*., 54 (1996), 111-120.