2.6 Dual Spaces 3 - 2.6. Since T t f = f T and T (W ) W, f...

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2.6. Since T t f = fT and T ( W ) W, fT ( W ) f ( W ) = 0 T t f W T t ( W ) W W is T t -invariant ( ) If T ( W ) * W, w W s.t. T ( w ) W by the exercise 13, f W s.t. f ( T ( w )) 6 = 0 T t f ( w ) = fT ( w ) 6 = 0 i.e. f W s.t. T t f / W W is T -invariant 18. Φ : V * → L ( S,F ) (Actually L ( S,F ) ≡ L ( V,F )) (i) Clearly L ( S,F ) is a vector space over F and (ii) Φ is a linear map d f,g ∈ L ( S,F ) , Φ( f + g ) = ( f + g ) | s = f | s + g | s = Φ( f ) + Φ( g ) Φ( αf ) = αf | s = αf s = α Φ( f ) c (iii) f ker Φ , Φ( f ) = f s = 0 f S = ( spanS ) = V = { 0 } ker Φ = { 0 } 136 PNU-MATH
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2.6. Φ is one-to-one (iv) By the exercise 34 in section 2.1 ( f s : S F, ! f : V F a linear map s.t. f ( x ) = f s ( x ) , x S ) f s ∈ L ( S,F ) , ! f V * s.t. Φ( f ) = f s Φ is onto 19. (i) Choose
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This note was uploaded on 02/14/2012 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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2.6 Dual Spaces 3 - 2.6. Since T t f = f T and T (W ) W, f...

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