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Unformatted text preview: ( , ) is complete with this norm Let ( ) be a Cauchy sequence in ( , ). For every x ( x ) ( x ) x Therefore ( ( x )) is a Cauchy sequence in , which converges since is complete. Define the function : by ( x ) = lim ( x ). is linear since ( x 1 + x 2 ) = lim ( x 1 + x 2 ) = lim ( x 1 ) + lim ( x 2 ) = ( x 1 ) + ( x 2 ) and ( x ) = lim ( x ) = lim ( x ) = ( x ) To show that is bounded, we observe that ( x ) = lim ( x ) = lim ( x ) sup ( x ) sup x Since ( ) is a Cauchy sequence, ( ) is bounded (Exercise 1.100), that is there exists such that...
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.
- Fall '10