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Unformatted text preview: 2.83 By Proposition 2.3, ( ) is connected and hence an interval (Exercise 1.95). 2.84 The range ( ) is a compact subset of (Proposition 2.3). Therefore is bounded (Proposition 1.1). 2.85 Let ( ) denote the set of all continuous (not necessarily bounded) functionals on . Then ( ) = ( ) ( ) ( ), ( ) are a linear subspaces of the set of all functionals ( ) (Exercises 2.11, 2.78 respectively). Therefore ( ) = ( ) ( ) is a subspace of ( ) (Exercise 1.130). Clearly ( ) ( ). Therefore ( ) is a linear subspace of ( ). Let be a bounded function in the closure of ( ), that is ( ). We show that is continuous. For any > 0, there exists ( ) such that < / 3....
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- Fall '10