Operations on Subspaces in Real Unitary Space

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OPERATIONS ON SUBSPACES IN REAL UNITARY SPACE 2 (6) 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 ) = (W 1 +W 2 ) +W 3 . (7) Let V be a real unitary space and W 1 , W 2 be subspaces of V . Then W 1 is a subspace of W 1 +W 2 and W 2 is a subspace of W 1 +W 2 . (8) Let V be a real unitary space, W 1 be a subspace of V , and W 2 be a strict subspace of V . Then W 1 is a subspace of W 2 if and only if W 1 +W 2 = W 2 . (9) For every real unitary space V and for every strict subspace W of V holds 0 V +W = W and W + 0 V = W. (10) Let V be a real unitary space. Then 0 V + Ω V = the unitary space structure of V and Ω V + 0 V = the unitary space structure of V . (11) Let V be a real unitary space and W be a subspace of V . Then Ω V +W = the unitary space structure of V and W + Ω V = the unitary space structure of V . (12) For every strict real unitary space V holds Ω V + Ω V = V. (13) For every real unitary space V and for every strict subspace W of V holds W ∩W = W. (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 . (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 ) = (W 1 ∩W 2 ) ∩W 3 . (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 W 1 and W 1 ∩W 2 is a subspace of W 2 . (17) Let V be a real unitary space, W 2 be a subspace of V , and W 1 be a strict subspace of V . Then W 1 is a subspace of W 2 if and only if W 1 ∩W 2 = W 1 . (18) For every real unitary space V and for every subspace W of V holds 0 V ∩ W = 0 V and W ∩ 0 V = 0 V . (19) For every real unitary space V holds 0 V ∩ Ω V = 0 V and Ω V ∩ 0 V = 0 V . (20) For every real unitary space V and for every strict subspace W of V holds Ω V ∩W = W and W ∩ Ω V = W. (21) For every strict real unitary space V holds Ω V ∩ Ω V = V. (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 of W 1 +W 2 . (23) For every real unitary space V and for every subspace W 1 of V and for every strict subspace W 2 of V holds W 1 ∩W 2 +W 2 = W 2 . (24) For every real unitary space V and for every subspace W 1 of V and for every strict subspace W 2 of V holds W 2 ∩ (W 2 +W 1 ) = W 2 . (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 ∩ W 3 is a subspace of W 2 ∩ (W 1 +W 3 ). (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 , then W 2 ∩ (W 1 +W 3 ) = W 1 ∩W 2 +W 2 ∩W 3 . (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 is a subspace of (W 1 +W 2 ) ∩ (W 2 +W 3 ). (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 , then W 2 +W 1 ∩W 3 = (W 1 +W 2 ) ∩ (W 2 +W 3 ).
OPERATIONS ON SUBSPACES IN REAL UNITARY SPACE 3 (29) Let V be a real unitary space and W 1 , W 2 , W 3 be subspaces of V . If W 1 is a strict subspace of W 3 , then W 1 +W 2 ∩W 3 = (W 1 +W 2 ) ∩W 3 . (30) For every real unitary space V and for all strict subspaces W 1 , W 2 of V holds W 1 +W 2 = W 2 iff W 1 ∩W 2 = W 1 . (31) Let V be a real unitary space, W 1 be a subspace of V , and W 2 , W 3 be strict subspaces of V . If W 1 is a subspace of W 2 , then W 1 +W 3 is a subspace of W 2 +W 3 . (32) Let V be a real unitary space and W 1 , W 2 be subspaces of V . Then there exists a subspace W of V such that the carrier of W = (the carrier of W 1 ) ∪ (the carrier of W 2 ) if and only if W 1 is a subspace of W 2 or W 2 is a subspace of W 1 . 3. INTRODUCTION OF A SET OF SUBSPACES OF REAL UNITARY SPACE Let V be a real unitary space. The functor SubspacesV yielding a set is defined as follows: (Def. 3) For every set x holds x ∈ SubspacesV iff x is a strict subspace of V . Let V be a real unitary space. Note that SubspacesV is non empty. The following proposition is true (33) For every strict real unitary space V holds V ∈ SubspacesV. 4. DEFINITION OF THE DIRECT SUM AND LINEAR COMPLEMENT OF SUBSPACES Let V be a real unitary space and let W 1 , W 2 be subspaces of V . We say that V is the direct sum of W 1 and W 2 if and only if: (Def. 4) The unitary space structure of V = W 1 +W 2 and W 1 ∩W 2 = 0 V . Let V be a real unitary space and let W be a subspace of V . A subspace of V is called a linear complement of W if: (Def. 5) V is the direct sum of it and W. Let V be a real unitary space and let W be a subspace of V . Observe that there exists a linear complement of W which is strict. The following two propositions are true: (34) Let V be a real unitary space and W 1 , W 2 be subspaces of V . Suppose V is the direct sum of W 1 and W 2 . Then W 2 is a linear complement of W 1 . (35) Let V be a real unitary space, W be a subspace of V , and L be a linear complement of W. Then V is the direct sum of L and W and the direct sum of W and L. 5. THEOREMS CONCERNING THE SUM, LINEAR COMPLEMENT AND COSET OF SUBSPACE One can prove the following propositions: (36) Let V be a real unitary space, W be a subspace of V , and L be a linear complement of W. Then W + L = the unitary space structure of V and L +W = the unitary space structure of V . (37) Let V be a real unitary space, W be a subspace of V , and L be a linear complement of W. Then W ∩ L = 0 V and L ∩W = 0 V . (38) Let V be a real unitary space and W 1 , W 2 be subspaces of V . If V is the direct sum of W 1 and W 2 , then V is the direct sum of W 2 and W 1 . (39) Every real unitary space V is the direct sum of 0 V and Ω V and the direct sum of Ω V and 0 V .
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