The determination of minimal projections and extensions in $L\sp 1$
Bruce L. Chalmers, FREDERIC T. METCALF · Transactions of the American Mathematical Society · 1992
Equations are derived which are shown to be necessary and sufficient for finite rank projections in ${L^1}$ to be minimal. More generally, these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace of ${L^1}$ and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the ${L^1}$ setting. These equations are solved in terms of an ${L^1}$ best approximation problem and the required orthogonality conditions. Moreover, this solution has a simple geometric interpretation. Questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions, including the minimal ${L^1}$ projection onto the quadratics.