Linear Functionals

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 $T$.