Convex Regions and Projections in Minkowski Spaces

Abstract

By a well known theorem of Hahn-Banach every linear functional of norm M defined over a closed linear subspace of a Banach space' can be extended linearly to the entire space without increasing its norm. For operations the corresponding problem takes the following form: Given a linear operation u(x) whose domain of definition is a closed linear subspace of a Banach space B, and whose range lies in a Banach space B2, does there exist an operation U(x) defined over B1, with range in B2 and which coincide with u(x) over the subspace? How small can the norm of the extended operation be made? In the particular case where u(x) = x, B1 = domain of definition of u; the existence of an extension U(x) is equivalent to the existence of a complementary subspace or, as F. J. Murray2 has shown, to the existence of a projection of B, on the subspace. Conversely, if A(X) is such a projection and u(x) any linear operation, the linear operation U(X) = u(A(X)) is an extension of u(x) and its norm is < 11 u 11.11 A 11. The question of extension is thus equivalent to the discussion of best projections, i.e. of projections of least norm. It is relatively easy to exhibit examples of Banach spaces for which certain closed subspaces have no projections; even more, F. J. Murray3 has shown that this occurs within the function spaces L, and the sequence spaces lp! For such subspaces an extension of an operation is not always possible. If the Banach space is finite dimensional, i.e. if we are dealing with a Minkowski space, of n dimensions say, the existence of projections is trivial but the discussion of the best possible projections (i.e. those with a minimal norm) is interesting and may help to explain why, in certain infinitely dimensional spaces, projections fail to exist. We give in this paper a complete answer to the question of best possible projections in the case where the dimensionality of the subspace on which we project is by 1 less than the dimensionality of the entire space. The results obtained lead to some interesting theorems on convex regions; they are considered in section 6.