ON MATRIX-CLOSED FAMILIES SUBSPACES OF A BANACH-SPACE

Matrix-closed families of subspaces of a Banach space have been of importance in connection with generalizations of the local reflexivity theorem. The present paper provides sufficient conditions for a family of subspaces to be matrix-closed which can easily be checked. Using this result it is, e.g., simple to decide whether a family consisting of three subspaces is matrix-closed or not. It is also shown that this is in a sense the best possible general assertion, since no similar result can hold for the case of four subspaces.