A non-Archimedean inner product

We define an inner product on a vector space X over a non-Archimedean valued field K. Our goal is to penetrate non-Archimedean normed spaces by means of Hilbert space type arguments. The basic idea in defining a "non-Archimedean inner product" is to substitute the Cauchy-Schwarz inequality for conjugate linearity in the second argument. Non-Arch imedean inner products induce norms in the usual way. We take 'x normal to y' to mean < y, x > = 0 [note the reversal of order] and show that x normal to y implies that x is orthogonal to y, in the usual non-Archimedean sense. The converse is generally false. For certain Banach spaces X, there is always an inner product that generates the original norm. We characterize them in Theorem 4.1. For normed subspaces X of (c(0) (T), parallel to(.)parallel to(infinity)) over a non-Archimedean valued field K with formally real residue class field, we define the "symmetric inner product" of x = (x(t)) and y = (y(t)) to be Sigma(t is an element of T)x(t)y(t). This one is attractive because < x, y > = < y, x > for all x and y. In this context, we develop a Gram-Schmidt procedure to convert linearly independent sequences into "orthonormal" ones, discuss conditions under which an orthonormal sequence can be extended to a basis, investigate "normal complements" and decompositions and, finally, the special properties of linear maps A : c(0) -> c(0) which preserve normality.