In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual in order to obtain and different representations elements space provided by frame. relate introduced with classical one Hilbert apply it three types spaces: conjugate closed subfields complex numbers (in particular, cyclotomic fields), metric spaces, ultrametric normed...

In this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-Archimedean number with $alpha^{-2}neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic $alpha$-functional equation (0.1) in non-Archimedean Banach spaces.