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

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

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

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 ?
Solution

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