In semi-infinite linear optimization theory, one minimizes <p,v> subject to <B(t),v> ≥for all, where T is usually a compact Hausdorff space. Introducing a new concept of certain pseudocompactness, this paper deals with semi-infinite problems on noncompact spaces T. The Kuhn-Tucker-condition and the Kolmogoroff-criterion are derived and a characterization for strong unicity is given. The problem of best approximation is an application of this theory as is shown in the last part.