Duality for Normed Linear Spaces

Verify the two direct substitution assertions in the proof of Proposition 4.

Proposition 4: Let $X$ be a linear space, the functional $\psi$ belong to $X^{\sharp }$, and $\left\{ \psi_{i}\right\}_{i=1}^{n}$ be contained in $X^{\sharp }$. Then $\psi$ is a linear combination of $\left\{ \psi_{i}\right\}_{i=1}^{n}$ if and only if $\bigcap_{i=1}^{n} \ker \psi_{i} \subseteq \ker \psi$. The proof argues by induction on $n$. In the base case $n=1$, having chosen $x_{0} \neq 0$ with $\psi (x_{0}) = 1$ (so $\psi_{1}(x_{0}) \neq 0$) and defined $\lambda_{1} = 1/\psi_{1}(x_{0})$, a direct substitution shows $\psi = \lambda_{1}\psi_{1}$. In the inductive step, given real numbers $\lambda_{1}, \ldots , \lambda_{k-1}$ for which $\psi = \sum_{i=1}^{k-1} \lambda_{i} \psi_{i}$ on $Y = \ker \psi_{k}$, and $x_{0}$ with $\psi_{k}(x_{0}) = 1$, one defines $\lambda_{k} = \psi (x_{0}) - \sum_{i=1}^{k-1} \lambda_{i} \psi_{i}(x_{0})$; a direct substitution shows $\psi = \sum_{i=1}^{k} \lambda_{i} \psi_{i}$ on $X$.

Let $X_{0}$ be a codimension 1 subspace of a normed linear space $X$. Show that $X_{0}$ is closed with respect to the strong topology if and only if $X_{0} = \operatorname {ker} \psi$ for some $\psi \in X^{*}$.

Show that if $X$ is a finite dimensional normed linear space, then every linear functional on $X$ is continuous.

Let $X$ be a finite dimensional normed linear space of dimension $n$. Let $\left\{ e_{1}, \ldots , e_{n}\right\}$ be a basis for $X$. For $1 \leq i \leq n$, define $\psi_{i} \in X^{*}$ by $\psi_{i}(x) = x_{i}$ for $x = x_{i} e_{i}+\cdots +x_{n} e_{n} \in X$. Show that $\left\{ \psi_{1}, \ldots , \psi_{n}\right\}$ is a basis for $X^{*}$. Thus $\operatorname {dim} X^{*} = n$.

Let $X$ be a finite dimensional linear space. Show that the weak and strong topologies on $X$ are the same.

Show that every non-empty weakly open subset of an infinite dimensional normed linear space is unbounded with respect to the norm.

Let $X$ be a finite dimensional space. Show that the natural embedding $J: X \rightarrow X^{* *}$ is one-to-one. Then use Problem 5 to show that $J: X \rightarrow X^{* *}$ is onto, so $X$ is reflexive.

For a vector $v \neq 0$ in Euclidean space $\mathbb {R}^{n}$, explicitly exhibit a linear functional $\psi : \mathbb {R}^{n} \rightarrow \mathbb {R}$ for which $\psi (v) = 1$.

For a sequence $\left\{ x_{n}\right\} \neq 0$ in $\ell^{2}$, explicitly exhibit a continuous linear functional $\psi : \ell^{2} \rightarrow \mathbb {R}$ for which $\psi \left(\left\{ x_{n}\right\} \right) = 1$.

For a function $f \neq 0$ in $L^{p}[a, b], 1 \leq p \leq \infty$, explicitly exhibit a continuous linear functional $\psi : L^{p}[a, b] \rightarrow \mathbb {R}$ for which $\psi (f) = 1$.

Consider $C[a, b]$ with the maximum norm. For a function $f \neq 0$ in $C[a, b]$, explicitly exhibit a continuous linear functional $\psi : C[a, b] \rightarrow \mathbb {R}$ for which $\psi (f) = 1$.

For $1 < p < \infty$, let $Y$ be a closed subspace of $L^{p}[a, b]$ of codimension 1. Show that there is function $g \in L^{q}[a, b]$, where $q$ is the conjugate of $p$, for which

Let $X$ be a normed linear space and $\psi$ belong to $X^{\sharp } \sim X^{*}$. Show that ker $\psi$ is dense, with respect to the strong topology, in $X$.

Let $X$ be the normed linear space of polynomials restricted to $[a, b]$. For $p \in X$, define $\psi (p)$ to be the sum of the coefficients of $p$. Show that $\psi$ is linear. Is $\psi$ continuous if $X$ has the topology induced by the maximum norm?

Let $X$ be the normed linear space of sequences of real numbers that have only a finite number of non-zero terms. For $x = \left\{ x_{n}\right\} \in X$, define $\psi (x) = \sum_{n = 1}^{\infty } x_{n}$. Show that $\psi$ is linear. Is $\psi$ continuous if $X$ has the topology induced by the $\ell^{\infty }$ norm?

Let $X$ be a linear space. A subset $E$ of $X$ is said to be linearly independent provided that each $x \in E$ fails to be a finite linear combination of points in $E \sim \left\{ x\right\}$. Define $\mathcal{F}$ to be the collection of non-empty subsets of $X$ that are linearly independent. Order $\mathcal{F}$ by set inclusion. Apply Zorn's Lemma to conclude that $X$ has a Hamel basis.

Provide an example of a discontinuous linear operator $T$ from a normed linear space $X$ to a normed linear space $Y$ for which $T$ has a closed graph. Does this contradict the Closed Graph 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

Show that a collection of bounded linear functionals is equicontinuous if and only if it is uniformly bounded.

Let $X$ be a separable normed linear space. Show that its closed unit sphere $S = \left\{ x \in X \mid \left\| x\right\| = 1\right\}$ also is separable.

Find a compact metric space $X$ for which $C(X)$, normed by the maximum norm, is reflexive.

Let $c_{0}$ be the subspace of $\ell^{\infty }$ consisting of sequences that converge to 0. Show that $c_{0}$ is a closed subspace of $\ell^{\infty }$ whose dual space is isomorphic to $\ell^{1}$. Conclude that $c_{0}$ is not reflexive and therefore neither is $\ell^{\infty }$.

For $1 \leq p \leq \infty$, show that the sequence space $\ell^{p}$ is reflexive if and only if $1 < p < \infty$. (For $p = \infty$, see the preceding problem.)

Consider the functional $\psi \in (C[-1,1])^{*}$ defined by

Show that $\psi$ fails to take a maximum on the closed unit ball of $C[-1,1]$. Use this to provide another proof that $C[-1,1]$ fails to be reflexive.

For $1 < p < \infty$, show that a bounded sequence in $\ell^{p}$ converges weakly if and only if it converges componentwise.

For $1 \leq p < \infty$ and $[a, b]$ a closed, bounded interval of real numbers, show that a bounded sequence $\left\{ f_{n}\right\}$ in $L^{p}[a, b]$ converges weakly to $f$ if and only if $\left\{ \int_{E} f_{n} d m\right\} \rightarrow \int_{E} f d m$ for every Lebesgue measurable subset $E$ of $[a, b]$.

For $[a, b]$ a closed, bounded interval of real numbers, show that if a sequence $\left\{ f_{n}\right\}$ in $C[a, b]$ converges weakly, then it converges pointwise.

For $X$ and $Y$ normed linear spaces and an operator $S \in \mathcal{L}(X, Y)$, define the adjoint of $S$, $S^{*} \in \mathcal{L}\left(Y^{*}, X^{*}\right)$ by

Let $X$ be a reflexive Banach space and $T: X \rightarrow X$ a linear operator. Show that $T$ belongs to $\mathcal{L}(X, X)$ if and only if whenever $\left\{ x_{n}\right\}$ converges weakly to $x,\left\{ T\left(x_{n}\right)\right\}$ converges weakly to $T(x)$.

Let $X$ be a normed linear space and $W$ be a subspace of $X^{*}$. Show that the $W$-weak topology on $X$ is Hausdorff if and only if $W$ separates points in $X$.

Let $X$ be a normed linear space and $\psi : X \rightarrow \mathbb {R}$ be linear. Show that $\psi$ is continuous with respect to the weak topology if and only if it is continuous with respect to the strong topology.

Let $X$ be a locally convex topological vector space and $\psi : X \rightarrow \mathbb {R}$ be linear. Show that $\psi$ is continuous if and only if it is continuous at the origin.

Let $X$ be a locally convex topological vector space and $\psi : X \rightarrow \mathbb {R}$ be linear. Show that $\psi$ is continuous if and only if there is a neighborhood $\mathcal{O}$ of the origin for which $f(\mathcal{O}) \neq \mathbb {R}$.

Let $X$ be a normed linear space and $W$ a subspace of $X^{*}$ that separates points. For any topological space $Z$, show that a mapping $f: Z \rightarrow X$ is continuous, where $X$ has the $W$-weak topology, if and only if $\psi \circ f: Z \rightarrow \mathbb {R}$ is continuous for all $\psi \in W$.

Show that the topology on a finite dimensional locally convex topological vector space is induced by a norm.

Let $X$ be a locally convex topological space. Show that the linear space $X^{\prime }$ of all linear continuous functionals $\psi : X \rightarrow \mathbb {R}$ also has a topology with respect to which it is a locally convex topological space on which, for each $x \in X$, the linear functional $\psi \mapsto \psi (x)$ is continuous.

Let $X$ and $Y$ be locally convex topological vector spaces and $T: X \rightarrow Y$ be linear, one-toone, and onto. Show that $T$ is a topological homeomorphism if and only if it maps a base at the origin for the topology on $X$ to a base at the origin for the topology on $Y$.

Let $X$ be a linear space and the function $\sigma : X \rightarrow [0, \infty )$ have the following properties: for all $u, v \in X$,

1. $\sigma (u+v) \leq \sigma (u)+\sigma (v)$;

2. $\sigma (u) = 0$ if and only if $u = 0$;

3. $\sigma (u) = \sigma (-u)$. Define $\rho (u, v) = \sigma (u-v)$. Show that $\rho$ is a metric on $X$.

Let $X$ be the linear space of all Lebesgue measurable functions on $[0,1]$. Define

For $0 < p < 1$, let $X$ be the linear space of all measurable (with respect to the Lebesgue measure $m$) real-valued functions on $[0,1]$ for which $\left|f\right|^{p}$ is integrable. Define

Let $\mathcal{S}$ be the space of all sequences of real numbers, and define

Show that $\sigma$ induces a metric on $\mathcal{S}$ for which $\mathcal{S}$ is a topological vector space with the metric topology. What is the most general continuous linear functional on $\mathcal{S}$ ?

For each natural number $n$, let $e_{n}$ denote the sequence in $\ell^{2}$ whose nth component is 1 and other components vanish. Define $E = \left\{ e_{n}+n \cdot e_{m} \mid n \text{ and } m \text{ any natural numbers}, m > n\right\}$. We showed that 0 is a point of closure of $E$ but no sequence in $E$ converges weakly to 0.

Consider the topological space $X = E \cup \left\{ 0\right\}$ with the weak topology. Find a function $f: X \rightarrow \mathbb {R}$ that fails to be continuous at 0 and yet has the property that whenever a sequence $\left\{ x_{n}\right\}$ in $E$ converges weakly to 0 , its image sequence $\left\{ f\left(x_{n}\right)\right\}$ converges to $f(0)$.

Find a subset of the plane $\mathbb {R}^{2}$ for which the origin is an internal point but not an interior point.

Let $X$ be a locally convex topological vector space and $V$ a convex, symmetric with respect to the origin (that is, $V = -V$) neighborhood of the origin. If $p_{V}$ is the gauge functional for $V$ and $\psi$ is a linear real-valued functional on $X$ such that $\psi \leq p_{V}$ on $X$, show that $\psi$ is continuous.

Let $X$ be a locally convex topological vector space, $Y$ a closed subspace of $X$, and $x_{0}$ belong to $X \sim Y$. Show that there is a continuous functional $\psi : X \rightarrow \mathbb {R}$ such that

Let $X$ be a normed linear space and $W$ a proper subspace of $X^{*}$ that separates points. Let $\psi$ belong to $X^{*} \sim W$. Show that ker $\psi$ is strongly closed and convex but not $W$-weakly closed.

Let $X$ be a normed linear space. Show that the closed unit ball $B^{*}$ of $X^{*}$ is weak-* closed.

Show that the Hyperplane Separation Theorem may be amended as follows: the point $x_{0}$ may be replaced by a convex set $K_{0}$ that is disjoint from $K$ and the conclusion is that $K$ and $K_{0}$ can be separated by a closed hyperplane if $K_{0}$ is compact. If $K_{0}$ is open, there is a continuous linear functional $\psi$ on $X$ for which $\psi \left(x_{0}\right) < \inf_{x \in K} \psi (x)$ for all $x_{0} \in K_{0}$.

Show that the weak topology on an infinite dimensional normed linear space is not first countable.

Show that every weakly compact subset of a normed linear space is bounded with respect to the norm.

Let $Y$ be a closed subspace of a reflexive Banach space $X$. For $x_{0} \in X \sim Y$, show that there is a point in $Y$ that is closest to $x_{0}$.

Let $X$ be normed linear space, $W$ a finite dimensional subspace of $X^{*}$ and $\psi$ a functional in $X^{*} \sim W$. Show that there is a vector $x \in X$ such that $\psi (x) \neq 0$ while $\varphi (x) = 0$ for all $\varphi$ in $W$.

Let $X$ be a normed linear space. Show that any dense subset of $B^{*} = \left\{ \psi \in X^{*} \mid \left\| \psi \right\| \leq 1\right\}$ separates points in $X$.

Complete the final part of the proof of Proposition 24.

Proposition 24: Let $K$ be a convex subset of a linear space $X$ that contains the origin as an internal point, and let $p_{K}$ be the gauge functional for $K$, defined by $p_{K}(x) = \inf \left\{ \lambda > 0 \mid x \in \lambda \cdot K\right\}$. Then $p_{K}$ is subadditive and positively homogeneous. The proof given establishes only subadditivity and leaves the proof of positive homogeneity — that $p_{K}(\lambda x) = \lambda \cdot p_{K}(x)$ for all $\lambda > 0$ and $x \in X$ — as the part to complete.

Find an example of a bounded subset $A$ of a normed linear space $X, \mathcal{F}$ a set of functionals in $X^{*}$ containing $\mathcal{F}_{0}$ as a dense subset of $\mathcal{F}$ (dense in the sense of the norm topology on $X^{*}$) such that $\mathcal{F}$ and $\mathcal{F}_{0}$ generate different weak topologies for $X$, but the same weak topology for $A$.

Find the extreme points of each of the following subsets of the plane $\mathbb {R}^{2}$ :

In each of the following, $B$ denotes the closed unit ball of a normed linear space $X$.

A norm on a linear space is said to be strictly convex provided that whenever $u$ and $v$ are distinct unit vectors and $0 < \lambda < 1$, then $\left\| \lambda u+(1-\lambda ) v\right\| < 1$. Show that the Euclidean norm on $\mathbb {R}^{n}$ and the usual norm on $L^{p}[a, b], 1 < p < \infty$ are strictly convex.