These are Ph.D level questions. Mostly belong to Banach space & Normed space. Please have me to have detailed solutions. Thanks!
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1. Let T : Rn ? Rm be a linear mapping represented by the m?—n matrix (aij) with respect to the standard bases in Rn,Rm. Compute the norm of T if: (a) Rn is equipped with the l1 norm and Rm with the l1 norm. (b) Rn and Rm are both equipped with the l1 norm. 2. Let X be a normed linear space with closed unit ball BX, B a Banach space, and T : X ? B a continuous injective linear map. Prove that if T(BX) is closed in B, X is complete. 3. Let B be a Banach space. We say that a linear mapping P : B ? B is a projection if P2 = P. Prove that the following statements are equivalent: (a) B = M ? N, and, (b) There exist bounded projections P : B ? M, Q : B ? N such that P + Q = I and PQ = QP = 0. 4. Let B,B1 be Banach spaces and T : B ? R(T) ? B1 a bounded linear operator. Prove that the following statements are equivalent: (a) T??1 : R(T) ? B is bounded, (b) There exists a constant c > 0 such that ?T(x)?B1 = c ?x?B for all x ? B, and, (c) K(T) = {0} and R(T) is closed in B1.5. Let B be a Banach space, X a normed linear space, and Tn : B ? X bounded linear operators such that limn Tn(x) = T(x) in X for x ? B. Prove : (a) There is a constant c > 0 such that supn ?Tn? = c. (b) Prove that the conclusion in (a) follows provided that l(Tn(x)) ? l(T(x)) for all x ? B, l ? X, instead. (c) T : B ? X is a bounded linear operator and ?T? = lim infn ?Tn?. 1
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