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 5<br />
7. INTRODUCTION OF OPERATIONS ON SET OF SUBSPACES<br />
Let V be a real unitary space. The functor SubJo<strong>in</strong>V yield<strong>in</strong>g a b<strong>in</strong>ary operati<strong>on</strong> <strong>on</strong> <strong>Subspaces</strong>V is<br />
def<strong>in</strong>ed as follows:<br />
(Def. 7) For all elements A 1 , A 2 of <strong>Subspaces</strong>V and for all subspaces W 1 , W 2 of V such that A 1 = W 1<br />
and A 2 = W 2 holds (SubJo<strong>in</strong>V )(A 1 , A 2 ) = W 1 +W 2 .<br />
Let V be a real unitary space. The functor SubMeetV yield<strong>in</strong>g a b<strong>in</strong>ary operati<strong>on</strong> <strong>on</strong> <strong>Subspaces</strong>V<br />
is def<strong>in</strong>ed by:<br />
(Def. 8) For all elements A 1 , A 2 of <strong>Subspaces</strong>V and for all subspaces W 1 , W 2 of V such that A 1 = W 1<br />
and A 2 = W 2 holds (SubMeetV )(A 1 , A 2 ) = W 1 ∩W 2 .<br />
One can prove the follow<strong>in</strong>g propositi<strong>on</strong><br />
8. THEOREMS OF FUNCTIONS SUBJOIN, SUBMEET<br />
(54) For every real unitary space V holds 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is a lattice.<br />
Let V be a real unitary space. One can check that 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is latticelike.<br />
The follow<strong>in</strong>g propositi<strong>on</strong>s are true:<br />
(55) For every real unitary space V holds 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is lowerbounded.<br />
(56) For every real unitary space V holds 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is upperbounded.<br />
(57) For every real unitary space V holds 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is a bound lattice.<br />
(58) For every real unitary space V holds 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is modular.<br />
(59) For every real unitary space V holds 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is complemented.<br />
Let V be a real unitary space. Observe that 〈<strong>Subspaces</strong>V,SubJo<strong>in</strong>V,SubMeetV 〉 is lowerbounded,<br />
upper-bounded, modular, and complemented.<br />
We now state the propositi<strong>on</strong><br />
(60) Let V be a real unitary space and W 1 , W 2 , W 3 be strict subspaces of V . If W 1 is a subspace<br />
of W 2 , then W 1 ∩W 3 is a subspace of W 2 ∩W 3 .<br />
9. AUXILIARY THEOREMS IN REAL UNITARY SPACE<br />
One can prove the follow<strong>in</strong>g propositi<strong>on</strong>s:<br />
(61) Let V be a real unitary space and W be a strict subspace of V . Suppose that for every vector<br />
v of V holds v ∈ W. Then W = the unitary space structure of V .<br />
(62) Let V be a real unitary space, W be a subspace of V , and v be a vector of V . Then there<br />
exists a coset C of W such that v ∈ C.<br />
(63) Let V be a real unitary space, W be a subspace of V , v be a vector of V , and x be a set. Then<br />
x ∈ v +W if and <strong>on</strong>ly if there exists a vector u of V such that u ∈ W and x = v + u.