Miscellany1 - by choosing an orthonormal basis { e 1 , . ....

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Analysis 2 Miscellaneous Problems 1 1. Let H be a Hilbert space, in which v is a point and X a closed subspace. Prove that min { k v - x k : x X } = max { |h v | y i| : y X , k y k = 1 } . 2. Fix a 2 . Prove that the following subset of 2 is compact: A = { z = ( z n ) n = 1 : n > 1 ⇒ | z n | 6 | a n |} . 3. Let X be an N -dimensional subspace of C [0 , 1] with inner product given by h f | g i = Z 1 0 f ( t ) g ( t )d t . Define K : [0 , 1] × [0 , 1] C
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Unformatted text preview: by choosing an orthonormal basis { e 1 , . . . , e N } for X and setting K ( s , t ) = N X n = 1 e n ( s ) e n ( t ) . Prove that K is choice-independent and has the (reproducing) property that if f X then Z 1 K ( s , t ) f ( t )d t = f ( s ) . 1...
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