▁~ ▁F 2 . ▁If ▁E 1 ▁ ⊥ ▁E 2 ▁and ▁F 1 ▁ ⊥ ▁F 2 , ▁then ▁ ▁E 1 ▁+ ▁E 2 ▁~ ▁F 1 ▁+ ▁F 2 . ▁Add itivity ▁would ▁not ▁generally ▁hold ▁if ▁one ▁were ▁to ▁require ▁unit ary ▁equivalence ▁in ▁the ▁definition ▁of ▁~ , ▁i . e . ▁if ▁we ▁say ▁E ▁is ▁equivalent ▁to ▁F ▁if ▁u * E u ▁= ▁F ▁for ▁some ▁unit ary ▁u . ▁ ▁The ▁sub spaces ▁belonging ▁to ▁M ▁are ▁partially ▁ordered ▁by ▁inclusion , ▁and ▁this ▁indu ces ▁a ▁partial ▁order ▁ ≤ ▁of ▁pro jections . ▁There ▁is ▁also ▁a ▁natural ▁partial ▁order ▁on ▁the ▁set ▁of ▁equivalence ▁classes ▁of ▁pro jections , ▁induced ▁by ▁the ▁partial ▁order ▁ ≤ ▁of ▁pro jections . ▁If ▁M ▁is ▁a ▁factor , ▁ ≤ ▁is ▁a ▁total ▁order ▁on ▁equivalence ▁classes ▁of ▁pro jections , ▁described ▁in ▁the ▁section ▁on ▁traces ▁below . ▁ ▁A ▁projection ▁( or ▁sub space ▁belonging ▁to ▁M ) ▁E ▁is ▁said ▁to ▁be ▁a ▁finite ▁projection ▁if ▁there ▁is ▁no ▁projection ▁F ▁< ▁E ▁( mean ing ▁F ▁ ≤ ▁E ▁ ▁and ▁F ▁ ≠ ▁E ) ▁that ▁is ▁equivalent ▁to ▁E . ▁ ▁For ▁example , ▁all ▁finite - dimensional ▁pro jections ▁( or ▁sub spaces ) ▁are ▁finite ▁( since ▁is omet ries ▁between ▁Hil bert ▁spaces ▁leave ▁the ▁dimension ▁fixed ), ▁but ▁the ▁identity ▁operator ▁on ▁an ▁infinite - dimensional ▁Hil bert ▁space ▁is ▁not ▁finite ▁in