Duality

If $Y$ is a subspace of a vector space $X$, show there is a subspace $Z$ such that $Y \cap Z=\left\{ 0\right\}$ and $Y \oplus Z=X$. (This means that every $x \in X$ can be written in a unique way as $x=y+z$, with $y \in Y$ and $z \in Z$.

Let $0<p<1$. Prove that (12.1) defines a metric on $L^{p}\left(0,1\right)$.

Formula (12.1): $d(f, g)=\int_{0}^{1}\left|f(t)-g(t)\right|^{p} \, d t$.

Prove that the extension of $T$ to $X$ in the proof of Proposition 12.1.10 is a linear functional.

Proposition 12.1.10: Let $Y$ be a linear subspace of a linear space $X$. Then each linear functional on $Y$ has an extension to a linear functional on $X$. (In particular, for each $x \in X \setminus \left\{ 0\right\}$, there is a linear functional $T$ such that $Tx \neq 0$.) In the proof: let $Z$ be a subspace of $X$ such that $X = Y \oplus Z$, let $T$ be a linear functional on $Y$, and extend $T$ to $X$ by defining $Tx = Ty$, where $x=y+z$ is the decomposition with $y \in Y, z \in Z$.

Find the norm of the linear functional on $\mathbb {R}^{2}$ defined by $T(x, y)=x-7 y$ if $\mathbb {R}^{2}$ is given the $\ell^{p}$ norm, $1 \leq p \leq \infty$.

Let $\delta$ be Dirac functional on $C[0,1]$ defined by $\delta (f)=f(0)$.

For what values of the constant $\alpha$ does

define a bounded linear functional on $C[0,1]$ ? For such $\alpha$ determine the norm of $T$.

For what values of the constant $\alpha$ does

define a bounded linear functional on $L^{p}\left(0,1\right)$ ? For such $\alpha$ determine the norm of $T$.

Let $X=C[0,1]$ with $\left\| \cdot \right\|_{\max }$ norm, let $g \in C[0,1]$, and define $T f= \int_{0}^{1} f(t) g(t) d t$. Prove that $T$ is a bounded linear functional on $C[0,1]$ and find its norm.

Let $X=C^{1}[0,1]$ with the norm $\left\| f\right\|_{(1)}:=\left\| f\right\|_{\infty }+\left\| f^{\prime }\right\|_{\infty }$, and define $T f=\int_{0}^{1} t f(t) d t$. Prove that $T$ is a bounded linear functional on $C^{1}[0,1]$ and find its norm.

Let $X=C[0,1]$ with $\left\| \cdot \right\|_{\max }$ norm, let $n \in \mathbb {N}$, let $\left\{ \alpha_{k}\right\}_{k=1}^{n}$ be real numbers, let $\left\{ x_{k}\right\}_{k=1}^{n} \subset [0,1]$, and define

Prove that $T$ is a bounded linear functional on $C[0,1]$ and find its norm. What if the sequence $\left\{ \alpha_{k}\right\}$ is infinite?

If $n \in \mathbb {N}$ let

Prove that $\left\| T_{n}\right\| =2$ for all $n \in \mathbb {N}$.

For a continuous function $f$ on $[0,2 \pi ]$, let $s_{n, x}(f)=s_{n}(f)(x)$, the $n$th partial sum of its Fourier series evaluated at $x \in [0,2 \pi ]$. Prove that $s_{n, x}$ is a bounded linear functional on $C[0,2 \pi ]$ and determine its norm.

Let $X=C[0,1]$ with $\left\| \cdot \right\|_{\max }$ norm, let $g \in L^{1}[0,1]$, and define $T f=\int_{0}^{1} f(t) g(t) d t$. Prove that $T$ is a bounded linear functional on $C[0,1]$ and find its norm.

Let $X$ be the space $\ell^{1}$ with the $\ell^{\infty }$ norm and let $T$ be a linear functional on $X$ defined by $T x=\sum_{k} x_{k}$. Prove or disprove: $T$ is continuous on $X$.

For $f \in C[0,1]$ let

For $x:=\left\{ x_{n}\right\} \in c_{0}$ let

For $x:=\left\{ x_{n}\right\} \in c$ let

For $x:=\left\{ x_{n}\right\} \in \ell^{1}$ let

Let $T$ and $T^{\prime }$ be two functionals on a vector space $X$ such that the kernel of $T$ is contained in the kernel of $T^{\prime }$. Prove that $T^{\prime }=c T$, for some real number $c$.

Let $X$ be a linear space, let $n \in \mathbb {N}$, and let $T_{1}, \ldots , T_{n}$ and $T$ be linear functionals on $X$ such that $\cap_{k=1}^{n} \operatorname {ker} T_{k} \subset \operatorname {ker} T$. Prove that $T$ is a linear combination of $T_{1}, \ldots , T_{n}$.

Prove that a linear functional on a normed space is bounded if and only if it maps bounded sets to bounded sets.

Let $X$ be a normed linear space and $x_{0} \in X$. Prove that a linear functional on a normed space is continuous if and only it is continuous at $x_{0}$.

Let $X$ be a normed linear space and $T: X \rightarrow \mathbb {R}$ a continuous additive map, i.e., $T(x+y)=T x+T y$, for all $x, y \in X$. Prove that $T$ is homogeneous, i.e., $T(\alpha x)=\alpha T x$, for all $\alpha \in \mathbb {R}$ and all $x \in X$.

Let $X$ be a normed linear space, and let $T$ be a bounded linear functional on $X$, and let $S$ be a set that is dense in $X$. Prove that if $T x=0$ for all $x \in S$ then $T=0$ on all of $X$.

Prove (12.6).

Inequality (12.6): Let $p>1$ and $q$ be conjugate numbers, let $\left\{ g_{n}\right\}_{n=1}^{\infty }$ be functions in $L^{q}\left(a, b\right)$ and $\left\{ c_{n}\right\}_{n=1}^{\infty }$ real numbers, and suppose that there is a function $f \in L^{p}\left(a, b\right)$ for which

Then for any sequence $\left\{ a_{n}\right\}$ of real numbers there exists $M>0$ so that

Let $X$ be a normed space and let $T$ be a nonzero bounded linear functional on $X$. Prove that for all $x \in X$,

Let $X$ be a normed linear space and $T$ a nonzero linear functional on $X$. Prove that there exists $y \in X$ such that every vector $x \in X$ can be written uniquely in the form $x=\lambda y+z$, where $\lambda \in \mathbb {R}$ and $z \in \operatorname {ker} T$.

Let $Y$ be a subspace of a vector space $X$ and let $T$ be a functional on $X$ such that $T(Y)$ is a proper subset of $\mathbb {R}$. Show that $T x=0$ for all $x \in Y$.

Let $X$ be a normed linear space and $T$ a nonzero linear functional on $X$. Prove that $T$ is continuous if and only if $\operatorname {ker} T^{C}$ has a nonempty interior.

Let $X$ be a normed linear space and $T$ a linear functional on $X$. Prove that $T$ is continuous if and only if there is an open set $G \subset X$ and a real number $\alpha$ such that $\alpha \notin T(G)$.

Let $X$ be a normed linear space and $T$ a linear functional on $X$. Prove that $T$ is continuous if and only if its kernel $\left\{ x \in X: T(x)=0\right\}$ is closed set.

Let $X$ be a normed linear space that is not finite-dimensional. Prove that there exists an unbounded linear functional on $X$.

Let $X$ be a normed linear space and $T$ a linear functional on $X$. Prove that the norm of $T$ is reciprocal to the distance in $X$ from 0 to the set $\left\{ x \in X: T x=1\right\}$.

Let $X$ be a normed linear space and $T$ a bounded linear functional on $X$. Suppose there exists a unit vector $x_{0} \in X$ such that $\left\| x_{0}-y\right\| \geq 1$ for every $y$ in the nullspace of $T$.

Let $\mathcal{M}=\left\{ (x, y) \in \mathbb {R}^{2}: 2 x-y=0\right\}$ and let $T: \mathcal{M} \rightarrow \mathbb {R}$ be defined by $f(x, y)=x$. Show that $T^{\prime }: \mathbb {R}^{2} \rightarrow \mathbb {R}$ defined by $T^{\prime }(x, y)=x / 5+2 y / 5$ is the unique Hahn-Banach extension of $T$.

Let $\mathcal{M}=\left\{ \left\{ x_{n}\right\} : x_{1}=x_{3}=x_{5}=\cdots =0\right\} \subset \ell^{1}$. Show that every non-zero functional $T: \mathcal{M} \rightarrow \mathbb {R}$ has infinitely many Hahn-Banach extensions to $\ell^{1}$.

Let $Y$ be the subspace of $C[a, b]$ consisting of all constant function and let $T \in Y^{*}$ be defined by $T f=f(a)$, for all $f \in Y$. Define

Prove that $T^{\prime }$ is a Hahn-Banach extension of $T$.

Prove that the function $T_{0}$ defined in the proof of the Hahn-Banach theorem is a well-defined, linear functional on the set $Z$ satisfying $\left\| T_{0}\right\|_{Z^{*}} \leq \left\| T\right\|_{Y^{*}}$.

The setting, from the proof of Theorem 12.2.2 (The Hahn-Banach Theorem: if $X$ is a normed linear space, $Y \subset X$ a linear subspace and $T$ a bounded linear functional on $Y$, then there is a linear functional $\tilde{T}$ on $X$ with $\tilde{T} x=T x$ for $x \in Y$ and $\left\| \tilde{T}\right\| =\left\| T\right\|$): let $\mathcal{A}$ be the collection of all pairs $(T^{\prime }, Y^{\prime })$, where $Y^{\prime } \supset Y$ is a linear subspace of $X$ and $T^{\prime }$ is an extension of $T$ to $Y^{\prime }$ that does not increase the norm, partially ordered by $(T^{\prime }, Y^{\prime }) \prec (T^{\prime \prime }, Y^{\prime \prime })$ if $Y^{\prime } \subset Y^{\prime \prime }$ and $T^{\prime \prime }=T^{\prime }$ on $Y^{\prime }$. Let $\mathcal{A}_{0}$ be a totally ordered subfamily of $\mathcal{A}$ and $Z=\cup \left\{ Y^{\prime }: (T^{\prime }, Y^{\prime }) \in \mathcal{A}_{0}\right\}$. For $z \in Z$ choose $(T^{\prime }, Y^{\prime }) \in \mathcal{A}_{0}$ with $z \in Y^{\prime }$ and define $T_{0}(z)=T^{\prime }(z)$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=a_{0}$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=\sum_{k=0}^{\operatorname {deg} p} a_{k}$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=\sum_{k=0}^{\operatorname {deg} p}(-1)^{k} a_{k}$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=\sum_{k=0}^{N} c_{k} a_{k}$, where $N$ is a fixed positive integer and $\left\{ c_{k}\right\}_{k=1}^{N}$ are fixed real numbers.

Let $X$ be a normed linear space and let $Y \subset X$ be a finite-dimensional linear subspace. Prove that there exists a closed subspace $Z \subset X$ such that $X=Y+Z$ and $Y \cap Z=\left\{ 0\right\}$.

Let $X$ be a normed linear space and let $Y \subset X$ be a finitedimensional linear subspace. Prove that there exists a vector $x_{0} \in X$ such that $d\left(x_{0}, Y\right)=1$.

Let $X$ be a normed linear space and $x, y$ two distinct points in $X$. Prove that there exists $T \in X^{*}$ for which $T x \neq T y$.

Let $V$ be the subspace of $L^{\infty }\left(0,1\right)$ defined by

Prove that there is $\varphi \in \left(L^{\infty }\left(0,1\right)\right)^{*}$ such that $\varphi (f)=\lim_{n} n \int_{[0,1 / n]} f(x) d x$ for every $f \in V$.

Prove that there exists a bounded linear functional LIM on $\ell^{\infty }$ with the following properties:

Let $X \neq 0$ and $Y$ be normed spaces for which $\mathcal{L}(X, Y)$ is a Banach space. Prove that $Y$ is also a Banach space.

Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ a linear operator such that $\varphi \circ T \in X^{*}$ for every $\varphi \in Y^{*}$. Prove that $T$ is bounded.

Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ an isometric isomorphism. Prove that there exists an isometric isomorphism $S: X^{*} \rightarrow Y^{*}$.

Prove Corollary 12.2.8.

Corollary 12.2.8: A subspace $Y$ of a Banach space $X$ is dense in $X$ if and only if every bounded linear functional that vanishes on $Y$ must vanish on $X$.

Let $X$ be a normed linear space, let $Y \subset X$ be a closed linear subspace, let $z \in X \backslash Y$, and let $V=\left\{ y+\lambda z: y \in Y, \lambda \in \mathbb {R}\right\}$. Prove that $V$ is a closed subspace of $X$.

Let $X$ be a normed linear space, let $Y \subset X$ be a linear subspace, and let $x_{0} \in X \backslash Y$. Prove that

Let $X, Y$ be normed spaces, let $X_{0}$ a closed subspace of $X$, let $a \in X \backslash X_{0}$, and let $b \in Y$. Prove that there is $T \in \mathcal{L}(X, Y)$ that vanishes on $X_{0}$ and such that $T a=b$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space, let $Y$ be a Banach space, and let $A: X \rightarrow Y$ be such that $T \circ A$ is integrable for every $T \in Y^{*}$. Define $L: Y^{*} \rightarrow \mathbb {R}$ by $L(T)=\int_{X} T \circ A d \mu$. Prove that $L \in Y^{* *}$.

Let $S$ be a subset of a normed linear space $X$. Prove that $S$ is bounded if and only if $T(S)$ is bounded for every $T \in X^{*}$.

Let $X$ be a normed linear space and $x_{0} \in X$. Prove that

Prove that every finite-dimensional normed linear space is reflexive.

Prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive.

Let $X$ be a reflexive normed space. Prove that if $Y$ is a closed subspace of $X$, then $Y$ is reflexive.

Let $X$ be a reflexive normed space and let $Y$ be a Banach space such that $Y^{*}$ is isometrically isomorphic to $X^{*}$. Prove that $Y$ is isometrically isomorphic to $X$.

Let $X$ be a reflexive normed space. Prove that every functional $T \in X^{*}$ attains its norm on $X$.

Let $X$ be a Banach space. If $\mathcal{M} \subset X$ and $\mathcal{N} \subset X^{*}$ are closed subspaces, let $\mathcal{M}^{0}=\left\{ T \in X^{*}: T \mid \mathcal{M}=0\right\}$ and $\mathcal{N}^{\perp }=\left\{ x \in X: T x=0 , \text{ for all } T \in \mathcal{N}\right\}$. Prove that:

Let $X$ be a normed space and $T, T_{1}, \ldots , T_{n} \in X^{*}$. Let $Y=\left\{ x \in X: T_{1}(x)=\cdots =T_{n}(x)=0\right\}$ and $S: Y \rightarrow \mathbb {R}$ be defined by $S(x)=T(x)$ on $Y$. If $\varphi$ is a bounded linear functional on $X$ that extends $S$ prove that there are $\alpha_{1}, \ldots , \alpha_{n} \in \mathbb {R}$ such that $\varphi =T+\alpha_{1} T_{1}+\cdots +\alpha_{n} T_{n}$.

Let $\mathcal{A}=\left\{ T \in \mathcal{L}\left(\ell^{2}\right): \left\langle T e_{1}, e_{1} \right\rangle -2 \left\langle T e_{1}, e_{2} \right\rangle =0\right\}$, and $\varphi$ a bounded linear functional on $\mathcal{A}$ defined by $\varphi (T)= \left\langle T e_{1}, e_{1} \right\rangle$. Prove that $\tilde{\varphi }: \mathcal{L}\left(\ell^{2}\right) \rightarrow \mathbb {R}$ defined by

is the unique Hahn-Banach extension of $\varphi$.

In Proposition 12.3.2 prove the case when $p>1$ and $\mu$ is $\sigma$-finite.

Proposition 12.3.2: Let $(X, \mathfrak {M}, \mu )$ be a $\sigma$-finite measure space, let $1 \leq p < \infty$, $1/p+1/q=1$, and let $f$ be an integrable function over $X$. Define a functional $T_{f}$ on $L^{\infty }\left(X, \mu \right)$ by $T_{f}(g) = \int_{X} fg \, d\mu$. If $T_{f}$ extends to a bounded linear functional on $L^{p}\left(X, \mu \right)$ then $f \in L^{q}\left(X, \mu \right)$ and $\left\| f\right\|_{q} \leq \left\| T_{f}\right\|$. (The case $p=1$ is proved in the text; only the case $p>1$, with $\mu$ $\sigma$-finite, remains.)

Use Hahn-Banach Theorem to prove that the dual space of $L^{\infty }\left(0,1\right)$ cannot be identified with $L^{1}\left(0,1\right)$.

Give an example of a measure space ( $X, \mathfrak {M}, \mu$ ) for which the Riesz Representation Theorem does extend to the case $p=\infty$.

The purpose of this exercise is to establish the following result: Theorem 12.3.5 need not hold for $p=1$ if the measure space is not $\sigma$-finite. Let $X=(0,1)$ and let $\mathfrak {M}$ be the $\sigma$-algebra consisting of sets that are either finite or countable or their complements are such. Finally, let $\mu$ be the counting measure on $\mathfrak {M}$.

Prove or disprove each of the following statements: There is $T \in \left(L^{q}\left(0,1\right)\right)^{*}, 1 \leq q<\infty$ such that $T f=f^{\prime }(0)$ for all

Let $1 \leq p<\infty$ and let $D \subset L^{p}\left(\mathbb {R}\right)$ consists of all functions $f \in L^{p}\left(\mathbb {R}\right)$ with compact support. Prove that $D$ is dense in $L^{p}\left(\mathbb {R}\right)$. Prove that this is not true if $p=\infty$.

Let $p>1$. Give an example of a subspace $X$ that is dense in $L^{r}\left(\mathbb {R}\right)$, for all $1<r<p$, but not in $L^{p}\left(\mathbb {R}\right)$.

Let $p>1$. Give an example of a subspace $X$ that is dense in $L^{p}\left(\mathbb {R}\right)$, but not in $L^{r}\left(\mathbb {R}\right)$ for any $1<r<p$.

Let ( $X, \mathfrak {M}, \mu$ ) be a $\sigma$-finite measure space and $1<p<\infty$. Prove that $L^{p}\left(X, \mu \right)$ is reflexive.

Let $Y$ be a subspace of $X$. Prove that:

Prove that in the case $1<p<\infty$, Riesz Theorem extends to measures that are not $\sigma$-finite.