The Hahn-Banach Theorem

Let $X$ be a linear subspace of $C[0,1]$ that is closed as a subset of $L^{2}[0,1]$. Verify the following assertions to show that $X$ has finite dimension. The sequence $\left\{ f_{n}\right\}$ belongs to $X$.

Let $X$ be a normed linear space, $\psi$ belong to $X^{*}$, and $\left\{ \psi_{n}\right\}$ be in $X^{*}$. Show that if $\left\{ \psi_{n}\right\}$ converges weak-* to $\psi$, then

Let $X = \mathbb {R}^{n}$ be normed with the Euclidean norm, $Y$ a subspace of $X$, and $\psi : Y \rightarrow \mathbb {R}$ a linear functional. Define $Y^{\perp }$ to be the linear subspace of $\mathbb {R}^{n}$ consisting of vectors orthogonal to $Y$. Then $\mathbb {R}^{n} = Y \oplus Y^{\perp }$. For $x = y+y^{\prime }, y \in Y, y^{\prime } \in Y^{\perp }$, define $\psi (x) = \psi (y)$. Show that this properly defines $\psi \in \left(\mathbb {R}^{n}\right)^{*}$, is an extension of $\psi$ on $Y$, and has the same norm as $\left.\psi \right|_{Y}$.

Let $X = L^{p} = L^{p}[a, b], 1 < p < \infty$ and $m$ be Lebesgue measure. For $f \neq 0$ in $L^{p}$, define

Use Hölder's Inequality to show that $\psi \in \left(L^{p}\right)^{*},\left\| \psi \right\|_{*} = 1$ and $\psi (f) = \left\| f\right\|_{p}$.

For each point $x$ in a normed linear space $X$, show that

Let $X$ be a normed linear space and $Y$ a closed subspace of $X$. Show that for each $x_{0} \in X \sim Y$, there is a $\psi \in X^{*}$ such that

Let $Y$ be a linear subspace of a normed linear space $X$ and $z$ be a vector in $X$. Show that

Let $X$ be a vector space. A subset $C$ of $X$ is called a cone provided that $x+y \in C$ and $\lambda x \in C$ whenever $x, y$ belong to $C$ and $\lambda > 0$. Define a partial order in $X$ by defining $x \prec y$ to mean $y-x \in C$. A linear functional $f$ on $X$ is said to be positive (with respect to the cone $C$) provided that $f \geq 0$ on $C$. Let $Y$ be any subspace of $X$ with the property that for each $x \in X$ there is a $y \in Y$ with $x \prec y$. Suppose $f$ is a linear functional on $Y$ that is positive with respect to the cone $C \cap Y$. Show that $f$ may be extended to a linear functional on $X$ that is positive with respect to $C$.

Let $X_{0}$ be a subset of a metric space $X$. Use the Tietze Extension Theorem to show that every continuous real-valued function on $X_{0}$ has a continuous extension to all of $X$ if and only if $X_{0}$ is closed. Does this contradict Theorem 7? Theorem 7: Let $X_{0}$ be a linear subspace of a normed linear space $X$. Then each bounded linear functional $\psi$ on $X_{0}$ has an extension to a bounded linear functional on all of $X$ that has the same norm as $\psi$. In particular, for each $x \in X$, $x \neq 0$, there is a $\psi \in X^{*}$ for which $\psi (x) = \left\| x\right\|$ and $\left\| \psi \right\|_{*} = 1$.

Let $F$ be closed subset of a metric space ($X, \rho$). Show that a point $x \in X$ belongs to $F$ if and only if every continuous real-valued function on $X$ that vanishes on $F$ also vanishes at $x$. Can this be used to prove Theorem 10? Theorem 10: Let $X_{0}$ be a subspace of the normed linear space $X$. Then a point $x$ in $X$ belongs to the closure of $X_{0}$ if and only if whenever a continuous functional $\psi \in X^{*}$ vanishes on $X_{0}$, it also vanishes at $x$.

Let $[a, b]$ be a closed, bounded interval of real numbers and consider $L^{\infty }[a, b]$, now formally considered as the collection of equivalence classes for the relation of pointwise equality almost everywhere among essentially bounded functions. Let $X$ be the subspace of $L^{\infty }[a, b]$ comprising those equivalence classes that contain a continuous function. Show that such an equivalence class contains exactly one continuous function. Thus $X$ is linearly isomorphic to $C[a, b]$ and therefore, modulo this identification, we may consider $C[a, b]$ to be a linear subspace $L^{\infty }[a, b]$. Show that $C[a, b]$ is a closed subspace of the Banach space $L^{\infty }[a, b]$.

Define $\psi : C[a, b] \rightarrow \mathbb {R}$ by $\psi (f) = f(a)$ for all $f \in C[a, b]$. Use Theorem 7 to extend $\psi$ to a continuous linear functional on all of $L^{\infty }[a, b]$ (see the preceding problem). Theorem 7: Let $X_{0}$ be a linear subspace of a normed linear space $X$. Then each bounded linear functional $\psi$ on $X_{0}$ has an extension to a bounded linear functional on all of $X$ that has the same norm as $\psi$. In particular, for each $x \in X$, $x \neq 0$, there is a $\psi \in X^{*}$ for which $\psi (x) = \left\| x\right\|$ and $\left\| \psi \right\|_{*} = 1$. Show that there is no function $h \in L^{1}[a, b]$ for which