## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2144

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

show that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

**topology**of X * , and is bounded . It remains only toshow that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

Page 2203

The mapping E HO ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

The mapping E HO ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

**topology**in 1 . To see this , note that sets of the form { a | | ( T ...Page 2278

If E * € & * we define m ( E * ) , the multiplicity of E * , to be the smallest cardinal

power of a set A of vectors such that E * X * is spanned in the X -

manifolds { N ( x * ) / 2 * € A } . The next lemma establishes the required continuity

...

If E * € & * we define m ( E * ) , the multiplicity of E * , to be the smallest cardinal

power of a set A of vectors such that E * X * is spanned in the X -

**topology**by themanifolds { N ( x * ) / 2 * € A } . The next lemma establishes the required continuity

...

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