hw4 - Mathematics Department Stanford University Math 175...

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Mathematics Department Stanford University Math 175 Homework 4 1. Suppose that H is a Hilbert space and { x k } k D 1 ; 2 ;::: is a sequence in H with the property that { .x k ;x/ } k D 1 ; 2 ;::: is bounded for each given x 2 H . (i) If S j D { x 2 H W j .x k ;x/ j ± j 8 k D 1 ; 2 ;::: } , j D 1 ; 2 ;::: , prove that each S j is a closed convex subset of H and [ 1 j D 1 S j D H . (ii) If the sets S j are as in (i), prove that some S j contains a ball. Hint: hw2, Q.4 2. Suppose that H is a Hilbert space and { x k } k D 1 ; 2 ;::: is a sequence in H with the property that { .x k ;x/ } k D 1 ; 2 ;::: is bounded for each given x 2 H . Prove that { x k } k D 1 ; 2 ;::: is a bounded sequence in H (i.e. there is M > 0 such that k x k k < M 8 k ). (!) Note: This property is called the uniform boundedness principle. 3. Let X be a complex normed linear space and suppose that f W X ! C is linear (“ f is a linear functional on X ”). (i) Prove that ker
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This note was uploaded on 02/01/2011 for the course MATH 171 taught by Professor R during the Fall '09 term at Stanford.

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