Linear Functionals

Suppose $V$ is a normed vector space and $\varphi$ is a linear functional on $V$. Suppose $\alpha \in \mathbf{F} \setminus \left\{ 0\right\}$. Prove that the following are equivalent:

1. $\varphi$ is a bounded linear functional.

2. $\varphi^{-1}(\alpha )$ is a closed subset of $V$.

3. $\overline{\varphi^{-1}(\alpha )} \neq V$.

Suppose $\varphi$ is a linear functional on a vector space $V$. Prove that if $U$ is a subspace of $V$ such that $\operatorname {null}\varphi \subseteq U$, then $U = \operatorname {null}\varphi$ or $U = V$.

Suppose $\varphi$ and $\psi$ are linear functionals on the same vector space. Prove that

if and only if there exists $\alpha \in \mathbf{F}$ such that $\psi = \alpha \varphi$.

For this exercise and the next, $\mathbf{F}^{n}$ should be endowed with the norm $\left\| \cdot \right\|_{\infty }$ as defined in Example 6.34.

Suppose $n \in \mathbb {N}$ and $V$ is a normed vector space. Prove that every linear map from $\mathbf{F}^{n}$ to $V$ is continuous.

For this exercise, as for the previous one, $\mathbf{F}^{n}$ should be endowed with the norm $\left\| \cdot \right\|_{\infty }$ as defined in Example 6.34.

Suppose $n \in \mathbb {N}$, $V$ is a normed vector space, and $T: \mathbf{F}^{n} \rightarrow V$ is a linear map that is one-to-one and onto $V$.

Suppose $n \in \mathbb {N}$.

Suppose $V$ and $W$ are normed vector spaces and $V$ is finite-dimensional. Prove that every linear map from $V$ to $W$ is continuous.

Prove that every finite-dimensional normed vector space is a Banach space.

Prove that every finite-dimensional subspace of each normed vector space is closed.

Give a concrete example of an infinite-dimensional normed vector space and a basis of that normed vector space.

Show that the collection $\mathcal{A} = \left\{ k \mathbb {Z}: k = 2,3,4, \ldots \right\}$ of subsets of $\mathbb {Z}$ satisfies the hypothesis of Zorn's Lemma (6.60).

Prove that every linearly independent family in a vector space can be extended to a basis of the vector space.

Suppose $V$ is a normed vector space, $U$ is a subspace of $V$, and $\psi : U \rightarrow \mathbb {R}$ is a bounded linear functional. Prove that $\psi$ has a unique extension to a bounded linear functional $\varphi$ on $V$ with $\left\| \varphi \right\| = \left\| \psi \right\|$ if and only if

for every $h \in V \setminus U$.

Show that there exists a linear functional $\varphi : \ell^{\infty } \rightarrow \mathbf{F}$ such that

for all $\left(a_{1}, a_{2}, \ldots \right) \in \ell^{\infty }$ and

for all $\left(a_{1}, a_{2}, \ldots \right) \in \ell^{\infty }$ such that the limit above on the right exists.

Suppose $B$ is an open ball in a normed vector space $V$ such that $0 \notin B$. Prove that there exists $\varphi \in V^{\prime }$ such that

for all $f \in B$.

Show that the dual space of each infinite-dimensional normed vector space is infinite-dimensional.

A normed vector space is called separable if it has a countable subset whose closure equals the whole space.

Suppose $V$ is a separable normed vector space. Explain how the Hahn-Banach Theorem (6.69) for $V$ can be proved without using any results (such as Zorn's Lemma) that depend upon the Axiom of Choice.

A normed vector space is called separable if it has a countable subset whose closure equals the whole space.

Suppose $V$ is a normed vector space such that the dual space $V^{\prime }$ is a separable Banach space. Prove that $V$ is separable.

A normed vector space is called separable if it has a countable subset whose closure equals the whole space.

Prove that the dual of the Banach space $C([0,1])$ is not separable; here the norm on $C([0,1])$ is defined by $\left\| f\right\| = \sup \left|f\right|$.

The double dual space of a normed vector space is defined to be the dual space of the dual space. If $V$ is a normed vector space, then the double dual space of $V$ is denoted by $V^{\prime \prime }$; thus $V^{\prime \prime } = \left(V^{\prime }\right)^{\prime }$. The norm on $V^{\prime \prime }$ is defined to be the norm it receives as the dual space of $V^{\prime }$.

Define $\Phi : V \rightarrow V^{\prime \prime }$ by

for $f \in V$ and $\varphi \in V^{\prime }$. Show that $\left\| \Phi f\right\| = \left\| f\right\|$ for every $f \in V$.

Suppose $V$ is an infinite-dimensional normed vector space. Show that there is a convex subset $U$ of $V$ such that $\overline{U} = V$ and such that the complement $V \setminus U$ is also a convex subset of $V$ with $\overline{V \setminus U} = V$.