By Arunava Mukherjea, K. Pothoven

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Suppose X is reflexive. To prove that the natural map J: X*** is onto, let x*** E X***. We define -+ x* = x*** 0 Jx , where Jx is the natural map from X onto X**. Then x* E X* and J(x*)[Jx(x») = Jx(x)[x*) = x*(x) = x***[Jx(x»). Since Jx(X) = X**, x*** = J(x*) and J is onto. To prove the converse, let X* be reflexive. If X is not reflexive, then Jx(X) is a closed proper subspace of X**. 3 there exists x*** E X*** such that x***(x**) *- 0 for some x** E X** - Jx(X) and x***[Jx(x») = 0 for each x EX.

For each gE L q , define Tg E Lp * by Tg(f) = fig dft. Show that for p > 1, I Tg I = I g Ilq, and that for p = 1 this equality holds for all g E Loo if and only if the measure is semifinite. 6. For any fixed l(t) E qo, 1] (under the uniform norm), let cP be the linear functional on C[O, I] defined by *- *- CP(g(t») = J:/(t)g(t) dt. Then show that cP is bounded and find I cP II. 7. Prove that 100 as well as Loo[O, I] is not separable. [Hint: Let Xk E 100 and Xk = (Xkl, Xk2, ... ). Define x = (aI' a 2, ...

Let (Yn) be a Cauchy sequence in Y. Then we can find a sequence (nk) of positive integers such that nk < nk+l and for each k, II Yn1:+1 - Yn1: \I < 1/2k. l:+1 - Ynlc II. Then L:'1 I Xk I < 00. Since X is complete, there exists x E X such that x = limn~L~_IXk. l:+1- Ynl + T(x). 1 Now as applications of the Open Mapping Theorem, we present the next three propositions and also the Closed Graph Theorem. 10. 11. Let X be a vector space that is complete in each of the norms I • " and I • Ill. Suppose there exists k> 0 such that I x I

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