Approximation in vectorial inner product spaces
by
João de Deus Marques
Centro de Matemática e Aplicações FCT-UNL

Of special interest among vectorially normed spaces are the vectorial inner product spaces, the theory of which is richer and retains many features of Euclidean spaces. In these spaces the vectorial norm p is defined in terms of a " vectorial inner product" F, in a natural way by the formula p(.)=(F(., .))^1/2. In this paper we establish the following result: Let (E, p(.)=(F(., .))^1/2) be a complete inner product space, p a regular vectorial norm, G a proper closed linear subspace of E. If the restriction of p to G is still regular then, given u ∈ E\G , there exists a unique element u^* ∈ G, such that p(u-u^*)=min_{w ∈ G} p(u-w).

Date received: May 28, 2008