The functions under approximation here have as a domain a finite dimensional Banach space with dimension \(N\in {\mathbb {N}}\) and are with values in \( {\mathbb {R}}^{N}\) . Exploiting some topological properties of the above we are able to perform Neural Network multivariate approximation to the above functions. The treatment is quantitative. We produce multivariate Jackson type inequalities involving the modulus of continuity of the function under approximation. The established convergences are pointwise and uniform. Perturbation and symmetrization to our operators lead to accelerated speeds of convergence.
