Let X and Y be normed spaces. We proved the following earlier in the course: If...

 

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Let X and Y be normed spaces. We proved the following earlier in the course: If Y is a Banach space, then B(X, Y) is a Banach space. In this exercise you will prove a converse statement. (a) Prove: If B(X, Y) is a Banach space and X {0}, then Y is a Banach space. (Hint: Choose to X {0}. Apply the Distance Attaining Functional Lemma when the subspace is {0} to obtain an f € X* with nice properties. If (yn) is any Cauchy sequence in Y, define a sequence of maps (Tn) by Tn(x) = f(x)yn.) (b) Is the implication in (a) true when X = {0}? Explain. Let X and Y be normed spaces. We proved the following earlier in the course: If Y is a Banach space, then B(X, Y) is a Banach space. In this exercise you will prove a converse statement. (a) Prove: If B(X, Y) is a Banach space and X {0}, then Y is a Banach space. (Hint: Choose to X {0}. Apply the Distance Attaining Functional Lemma when the subspace is {0} to obtain an f € X* with nice properties. If (yn) is any Cauchy sequence in Y, define a sequence of maps (Tn) by Tn(x) = f(x)yn.) (b) Is the implication in (a) true when X = {0}? Explain.

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a Prove If BXY is a Banach space and not equal 0 t hen Y is a Banach space Given that BXYBXY is a Ba... View the full answer