I'm struggling to follow a certain proof in Luenberger's book "Optimization by Vector Space Methods". If anyone could help shed some light on what I'm missing, I would certainly appreciate it. Let me start with a couple of definitions.
Definition 1 Let $S$ be a subset of a normed linear space $X$. The orthogonal complement of $S$, denoted $S^\perp$, consists of all elements $x^* \in X^*$ ($X^*$ denotes the continuous dual space) orthogonal to every vector in $S$.
Definition 2 Given a subset $U$ of the dual space $X^*$, we define the orthogonal complement of $U$ in $X$ as the set $^\perp U \subset X$ consisting of all elements in $X$ orthogonal to every vector in $U$.
Definition 3 The vectors $x \in X$ and $x^* \in X^*$ are said to be orthogonal if $\langle x, x^*\rangle =0$.
$\langle \cdot,\cdot \rangle$ denotes the dual pairing.
Theorem Let $M$ be a closed subspace of a normed space $X$. Then $^\perp[M^\perp] = M$.
Proof It is clear that $M \subset{}^\perp[M^\perp]$. To prove the converse, let $x \notin M$. On the subspace $[x + M]$ generated by $x$ and $M$, define the linear functional $f(\alpha x + m) = \alpha $ for $m \in M$. Then \begin{equation} \|\,f\|=\sup_{m\in M}\frac{f(x+m)}{\|x+m\|}=\frac{1}{\inf_{m}\|x+m\|} \end{equation} and since $M$ is closed, $\|\,f\| < \infty$. Thus by the Hahn-Banach theorem, we can extend $f$ to an $x^* \in X^*$. Since $f$ vanishes on M, we have $x^* \in M^\perp$ But also $\langle x, x^*\rangle = 1$ and thus $x \notin{}^\perp[M^\perp]$.
Comments I'm struggling to understand the necessity of the subspace being closed. Would bounded not suffice?
My understanding of the proof is as follows. Using the definition of the norm of an $f$ in the dual space and positivity we get the first equality in the displayed equation. Noticing that $\alpha=1$ and interchanging $\sup$ and $\inf$ by taking the $\inf$ into the denominator gives the second equality. I would have thought from here we would be able to guarantee finiteness of the norm. We then use the Hahn-Banach Theorem to extend $f$ to $x^* \in X^*$. Then using $\alpha=0$ implies $f=0$ on $M$. Thus $f(x_1)=\langle x^*,x_1\rangle=0$ where $x_1 \in M$ so they are orthogonal. However, $\alpha=1$ implies $f(x)=1=\langle x^*,x\rangle$ hence $x \notin{}^\perp[M^\perp]$.
If $M$ is not closed, then for each $x \in \overline{M} \backslash M$ there is a sequence $(m_n)$ in $M$ such that $\lVert x - m_n \rVert \to 0$ as $n \to \infty$. But then $\inf_m \lVert x + m \rVert = 0$ since $-m_n \in M$.
Note that no linear subspace except the trivial one (having $0$ as its only element) is bounded.