Operations on Subspaces in Real Unitary Space
Contents - Markun Cs Shinshu U Ac Jp
Contents - Markun Cs Shinshu U Ac Jp
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OPERATIONS ON SUBSPACES IN REAL UNITARY SPACE 2<br />
(6) For every real unitary space V and for all subspaces W 1 , W 2 , W 3 of V holds W 1 + (W 2 +<br />
W 3 ) = (W 1 +W 2 ) +W 3 .<br />
(7) Let V be a real unitary space and W 1 , W 2 be subspaces of V . Then W 1 is a subspace of<br />
W 1 +W 2 and W 2 is a subspace of W 1 +W 2 .<br />
(8) Let V be a real unitary space, W 1 be a subspace of V , and W 2 be a strict subspace of V .<br />
Then W 1 is a subspace of W 2 if and <strong>on</strong>ly if W 1 +W 2 = W 2 .<br />
(9) For every real unitary space V and for every strict subspace W of V holds 0 V +W = W and<br />
W + 0 V = W.<br />
(10) Let V be a real unitary space. Then 0 V + Ω V = the unitary space structure of V and Ω V +<br />
0 V = the unitary space structure of V .<br />
(11) Let V be a real unitary space and W be a subspace of V . Then Ω V +W = the unitary space<br />
structure of V and W + Ω V = the unitary space structure of V .<br />
(12) For every strict real unitary space V holds Ω V + Ω V = V.<br />
(13) For every real unitary space V and for every strict subspace W of V holds W ∩W = W.<br />
(14) For every real unitary space V and for all subspaces W 1 , W 2 of V holds W 1 ∩W 2 = W 2 ∩W 1 .<br />
(15) For every real unitary space V and for all subspaces W 1 , W 2 , W 3 of V holds W 1 ∩(W 2 ∩W 3 ) =<br />
(W 1 ∩W 2 ) ∩W 3 .<br />
(16) Let V be a real unitary space and W 1 , W 2 be subspaces of V . Then W 1 ∩W 2 is a subspace of<br />
W 1 and W 1 ∩W 2 is a subspace of W 2 .<br />
(17) Let V be a real unitary space, W 2 be a subspace of V , and W 1 be a strict subspace of V .<br />
Then W 1 is a subspace of W 2 if and <strong>on</strong>ly if W 1 ∩W 2 = W 1 .<br />
(18) For every real unitary space V and for every subspace W of V holds 0 V ∩ W = 0 V and<br />
W ∩ 0 V = 0 V .<br />
(19) For every real unitary space V holds 0 V ∩ Ω V = 0 V and Ω V ∩ 0 V = 0 V .<br />
(20) For every real unitary space V and for every strict subspace W of V holds Ω V ∩W = W and<br />
W ∩ Ω V = W.<br />
(21) For every strict real unitary space V holds Ω V ∩ Ω V = V.<br />
(22) For every real unitary space V and for all subspaces W 1 , W 2 of V holds W 1 ∩W 2 is a subspace<br />
of W 1 +W 2 .<br />
(23) For every real unitary space V and for every subspace W 1 of V and for every strict subspace<br />
W 2 of V holds W 1 ∩W 2 +W 2 = W 2 .<br />
(24) For every real unitary space V and for every subspace W 1 of V and for every strict subspace<br />
W 2 of V holds W 2 ∩ (W 2 +W 1 ) = W 2 .<br />
(25) For every real unitary space V and for all subspaces W 1 , W 2 , W 3 of V holds W 1 ∩W 2 +W 2 ∩<br />
W 3 is a subspace of W 2 ∩ (W 1 +W 3 ).<br />
(26) Let V be a real unitary space and W 1 , W 2 , W 3 be subspaces of V . If W 1 is a subspace of W 2 ,<br />
then W 2 ∩ (W 1 +W 3 ) = W 1 ∩W 2 +W 2 ∩W 3 .<br />
(27) For every real unitary space V and for all subspaces W 1 , W 2 , W 3 of V holds W 2 +W 1 ∩W 3<br />
is a subspace of (W 1 +W 2 ) ∩ (W 2 +W 3 ).<br />
(28) Let V be a real unitary space and W 1 , W 2 , W 3 be subspaces of V . If W 1 is a subspace of W 2 ,<br />
then W 2 +W 1 ∩W 3 = (W 1 +W 2 ) ∩ (W 2 +W 3 ).
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