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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.
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2. Prove that a Banach space X is either reflexive or its second duals X^**,X^****,... are all distinct.
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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.
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4. A Banach space E is finite-dimensional if and only if each of its subspaces is closed.
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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.
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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.
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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.
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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).
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9. Are c and c_0 isometrically isomorphic ?
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10. Show that in a reflexive Banach space E, the closed unit ball is compact for the weak topology.
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