hw8 - Mathematics Department Stanford University Math 175...

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Unformatted text preview: Mathematics Department Stanford University Math 175 Homework 8 This is the last homework which will be graded; There will be a hw9, but that will not be graded 1. (a) H is a Hilbert space and { e n } n D 1 ; 2 ;::: , { f p } p D 1 ; 2 ;::: are complete orthonormal sequences in H . Prove that if a np D .e n ;f p / then P 1 p D 1 a np a mp D ı nm and P 1 n D 1 a np a nq D ı pq , where ı ij is the Kronecker delta. (b) If T 2 L .H;K/ where H;K are Hilbert spaces, and if { e n } ; { f p } are as in (a) prove that P n k T.e n / k 2 converges if and only if P p k T.f p / k 2 converges and in this case the two sums agree. Hint: Start by assuming P p k T.f p / k 2 converges, and, to avoid convergence difficulties, first consider the finite rank operator T N .x/ D P N p D 1 .x;f p /T.f p / instead of T . 2. (a) Suppose that for u 2 C 2 .OEa;bŁ/ (real valued twice continuously differentiable functions on OEa;bŁ ) we define Lu D .pu / C qu with p 2 C 1 .OEa;bŁ/;q 2 C .OEa;bŁ/ given and p > 0 everywhere in...
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