(Lax, Ch. 8, Exercise 3) Show that if $Y$ is a closed subspace of a normed linear space $X$, then ${Y}^{\ast}$ is isometrically isomorphic to the quotient of ${X}^{\ast}$ by the annihilator of $Y$in ${X}^{\ast}$. (The quotient is given the quotient topology.) At some point you need Hahn-Banach.

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.)

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 ${V}^{\ast \ast}$ onto $V$. (Hint: consider ${J}^{\ast}$, where $J$ is the canonical injection of $W$ into ${W}^{\ast \ast}$.)

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 $P:{\ell}^{\mathrm{\infty}}\to {c}_{0}$ 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 $\xi $supported 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 $Q{\xi}_{i}\ne 0$. (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.

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