Linear Functionals, Bounded Linear Functionals, and Weak Topologies

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?