HW #5 on Dual Spaces and Reflexivity

  • Due Oct 8, 2014 by 10am
  • Points 10
  1. (Lax, Ch. 8, Exercise 3) Show that if Y is a closed subspace of a normed linear space X, then Y^* is isometrically isomorphic to the quotient of X^* by the annihilator of Yin X^*. (The quotient is given the quotient topology.)  At some point you need Hahn-Banach.
  2. Fill in the details of the following point which was mentioned in class but skipped over quickly: If V is a Banach space and J is the canonical map V\to V^{\ast\ast}, show that J(V) is closed in V^{\ast\ast} in the norm topology. (Hint: The completeness of V is essential here, along with the fact that J is an isometry.)
  3. Show that if a Banach space  V is isometric to the dual of another Banach space W, then there is a norm-one projection P from V^{**} onto V. (Hint: consider J^*, where J is the canonical injection of W into W^{**}.)
  4. Prove that the Banach space c_0 of sequences tending to 0 is not the dual space of any other Banach space.  Here's a sketch.  Assume that c_0 is the dual of a Banach space V.  Then (by #2), there is a linear projection P\colon \ell^\infty\to c_0 with P^2=P and with norm 1.   Let Q=1-P and suppose you can show there is an infinite subset S of \mathbb{N} such that if \xi is supported in S, then Q\xi=0. Then P\xi=\xi, i.e., every \xisupported in S tends to 0, which is clearly false.  To construct S, first show (this is just combinatorics) that there are uncountably many infinite subsets S_i  of \mathbb{N} (i running over an uncountable index set I) with the property that any two of these have finite intersection.  Suppose you could find  \xi_i supported in S_i  for each i with Q\xi_i\ne0.  (Otherwise there is some S_i for which Q\xi=0 if \xi is supported in S_i.)  Then use the property of the S_i and take linear combinations of the \xi_i to contradict the fact that Q has norm 1.
1412776800 10/08/2014 10:00am