Many of the algebraic results discussed above survive the passage to this context. A given direct sum decomposition of formula_174 into complementary subspaces still specifies a projection, and vice versa. If formula_174 is the direct sum formula_178, then the operator defined by formula_179 is still a projection with range formula_42 and kernel formula_4. It is also clear that formula_6. Conversely, if formula_1 is projection on formula_174, i.e. formula_6, then it is easily verified that formula_186. In other words, formula_187 is also a projection. The relation formula_6 implies formula_189 and formula_174 is the direct sum formula_191.