Direct Sum

Def Sum of Subsets Let be subsets of , the sum of is the set of all sums of the elements in . That is .

Prop If the subsets are subspaces, then the sum of these subsets is a subspace.

Def Direct Sum Given a collection of subspaces , is a direct sum if each element can be expressed as a sum in a unique way. And if so, we write .

e.g. is the direct sum of the - plane and -axis.