MATH806 Midterm: MATH 806 Solutions2Exams
31 Jan 2019
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October 19: (10 points) let ( n)n 1 be an orthonormal sequence in an in nite dimensional hilbert space. (a) state bessel"s inequality. If h is a hilbert space and ( n)n 1 is an orthonormal sequence, Xn=1 (where ( , ) and k k are the inner product in h and the associated norm). (b) de ne what we mean when we say that ( n)n 1 is a complete orthonormal sequence. One possible de nition: the set cannot be extended to a larger orthonormal set. It becomes an equality for every x. (that is another equivalent: (10 points) prove that in an inner product space h kxk = sup. The result needs to be proved for x 6= 0, since it is straightforward for x = 0. = kxk, by the cauchy-schwarz inequality: (10 points) let (xn)n 1 be a sequence in a normed space x. Prove that if xn x, then xn x.
