As a part of the Approximation Theory and its Applications in Vector Spaces, approaches to achieve the Best Approximation by splines are reviewed in this research work, starting from the background of the linear spaces, followed by Hausdor locally convex spaces, the normed linear spaces and the particular case of Hilbert spaces. Special attention is given to the most signi cant Best Approximation Problems in Separated Locally Convex Spaces and to their straight connections with the Vector Optimization, that is, with the general Eciency. Remarkable consideration is given to these problems in H - locally convex spaces where the splines introduced by an original method represent the best approximation simultaneous and vectorial solutions as optimal interpolation elements. The survey covers illustrative references.