Unformatted text preview: K → H such that ( T ( x ) ,y ) K = ( x,S ( y )) H for every x ∈ H and y ∈ K, where ( , ) H and ( , ) K denote the inner products on H and K, respectively. Hint: for each ﬁxed y ∈ K, show that ( T ( x ) ,y ) K is a bounded linear functional on H, and think about applying the Riesz representation theorem. 4. For each f ∈ L 2 ([0 , 1]) , consider T ( f ) = g, where g ( t ) = R t f. Prove that T ( L 2 ([0 , 1]) ) ⊂ C 1 / 2 ([0 , 1]) where C 1 / 2 ([0 , 1]) is the space of H¨oldercontinuous functions g on [0 , 1] with exponent 1 / 2 , that is, the functions g : [0 , 1] → R such that there is a constant C with  g ( t 1 )g ( t 2 )  ≤ C  t 1t 2  1 / 2 ∀ t 1 ,t 2 ∈ [0 , 1] . 5 , 6 , 7 : Problems 7 . 1 , 7 . 2 , 7 . 4 of the text. 1...
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 Fall '09
 R
 Math, Linear Algebra, Hilbert space, Compact space, Linear functional, Riesz, trivial null space

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