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Problem 1
1. Let X and Y be Banach spaces and let T be a bounded linear map from X into Y. If T(X) is of the second category (in Y), then T(X)=Y.
Solution

Problem 2
2. Prove that a Banach space X is either reflexive or its second duals X^**,X^****,... are all distinct.
Solution

Problem 3
3. Let X be an infinite-dimensional Banach space. Show that there exists an infinite and strictly decreasing sequence (Y_n) of infinite-dimensional closed linear subspaces of X.
Solution

Problem 4
4. A Banach space E is finite-dimensional if and only if each of its subspaces is closed.
Solution

Problem 5
5. Let f be an entire function, and let T be a continuous linear mapping T: C -> C. Show that [f(T)]*=f(T*), where the stars denote the adjoints.
Solution

Problem 6
6. Prove that there exists a sequence of complex polynomials (P_n) such that lim P_n(0)=1 and lim P_n(z)=0 if z <> 0.
Solution

Problem 7
7. Prove the existence of a sequence of complex polynomials (P_n) such that for all Re(z) >= 0,lim P_n(z)=1 and for all 0>Re(z), lim P_n(z)=0.
Solution

Problem 8
8. Show that c* is isometrically isomorphic to l_1 (we work in the field of the complex numbers; c* denotes the complex vector space of the converging complex sequences).
Solution

Problem 9
9. Are c and c_0 isometrically isomorphic ?
Solution

Problem 10
10. Show that in a reflexive Banach space E, the closed unit ball is compact for the weak topology.
Solution


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