hw5 - K H such that ( T ( x ) ,y ) K = ( x,S ( y )) H for...

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Mathematics Department Stanford University Math 175 Homework 5 Due Thursday May 6. 1. Suppose X is a normed space and T : X X is a linear operator such that T ( B 1 (0)) is contained in a compact subset of X, and let I : X X be the identity map on X. Prove that ( I + T ) X is a closed linear subspace of X, provided I + T has trivial null space (i.e., x + T ( x ) 6 = 0 for x 6 = 0 ). Note: Actually the result here (that I + T has closed range) is true even without the additional hypothesis that I + T has trivial null space but the proof of that is a bit more difficult.) 2. For f L 2 C ([ - π,π ]) define T ( f ) = the complex sequence ( c 0 ,c 1 ,c - 1 ,...,c n ,c - n ,... ) , where c k = ( f,e k ) L 2 C (with e k = e ikt ) is the k -th Fourier coefficient of f . Prove that T is a linear isometry of L 2 C on l 2 , that is, ( Tf,Tg ) l 2 = ( f,g ) L 2 for all f,g L 2 . 3) Suppose that H,K are real Hilbert spaces and T : H K is a bounded linear map (i.e., T is linear and there is a constant C such that k T ( x ) k ≤ C k x k for all x H. ) Prove there is a bounded linear map S :
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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 xed 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 Holder-continuous 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 1-t 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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