Banach Spaces

Verify that each of the claimed metrics in Example 6.2 is indeed a metric.

Prove that every finite subset of a metric space is closed.

Prove that every closed ball in a metric space is closed.

Suppose $V$ is a metric space.

Suppose $V$ is a metric space.

Show that each sequence in a metric space has at most one limit.

Prove 6.9.

Prove that each open subset of a metric space $V$ is the union of some sequence of closed subsets of $V$.

Prove or give a counterexample: If $V$ is a metric space and $U, W$ are subsets of $V$, then $\overline{U} \cup \overline{W} = \overline{U \cup W}$.

Prove or give a counterexample: If $V$ is a metric space and $U, W$ are subsets of $V$, then $\overline{U} \cap \overline{W} = \overline{U \cap W}$.

Suppose $\left(U, d_{U}\right),\left(V, d_{V}\right)$, and $\left(W, d_{W}\right)$ are metric spaces. Suppose also that $T: U \rightarrow V$ and $S: V \rightarrow W$ are continuous functions.

Prove the parts of 6.11 that were not proved in the text.

Suppose a Cauchy sequence in a metric space has a convergent subsequence. Prove that the Cauchy sequence converges.

Verify that all five of the metric spaces in Example 6.2 are complete metric spaces.

Suppose $(U, d)$ is a metric space. Let $W$ denote the set of all Cauchy sequences of elements of $U$.

Show that if $a, b \in \mathbb {R}$ with $a+b i \neq 0$, then

Suppose $z \in \mathbb {C}$. Prove that

Suppose $z \in \mathbb {C}$. Prove that $\frac{\left|\operatorname {Re} z\right|+\left|\operatorname {Im} z\right|}{\sqrt{2}} \leq \left|z\right| \leq \left|\operatorname {Re} z\right|+\left|\operatorname {Im} z\right|$.

Suppose $w, z \in \mathbb {C}$. Prove that $\left|w z\right| = \left|w\right|\left|z\right|$ and $\left|w+z\right| \leq \left|w\right|+\left|z\right|$.

Suppose $(X, \mathcal{S})$ is a measurable space and $f: X \rightarrow \mathbb {C}$ is a complex-valued function. For conditions (b) and (c) below, identify $\mathbb {C}$ with $\mathbb {R}^{2}$. Prove that the following are equivalent:

1. $f$ is $\mathcal{S}$-measurable.

2. $f^{-1}(G) \in \mathcal{S}$ for every open set $G$ in $\mathbb {R}^{2}$.

3. $f^{-1}(B) \in \mathcal{S}$ for every Borel set $B \in \mathcal{B}_{2}$.

Suppose $(X, \mathcal{S})$ is a measurable space and $f, g: X \rightarrow \mathbb {C}$ are $\mathcal{S}$-measurable. Prove that

Suppose $(X, \mathcal{S})$ is a measurable space and $f_{1}, f_{2}, \ldots$ is a sequence of $\mathcal{S}$ measurable functions from $X$ to $\mathbb {C}$. Suppose $\lim_{k \rightarrow \infty } f_{k}(x)$ exists for each $x \in X$. Define $f: X \rightarrow \mathbb {C}$ by

Prove that $f$ is an $\mathcal{S}$-measurable function.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f: X \rightarrow \mathbb {C}$ is an $\mathcal{S}$-measurable function such that $\int \left|f\right| d \mu < \infty$. Prove that if $\alpha \in \mathbb {C}$, then

Suppose $V$ is a vector space. Show that the intersection of every collection of subspaces of $V$ is a subspace of $V$.

Suppose $V$ and $W$ are vector spaces. Define $V \times W$ by

Define addition and scalar multiplication on $V \times W$ by

Prove that $V \times W$ is a vector space with these operations.

Show that the map $f \mapsto \left\| f\right\|$ from a normed vector space $V$ to $\mathbf{F}$ is continuous (where the norm on $\mathbf{F}$ is the usual absolute value).

Prove that if $V$ is a normed vector space, $f \in V$, and $r > 0$, then

Show that the functions defined in the last two bullet points of Example 6.35 are not norms.

Prove that each Cauchy sequence in a normed vector space is bounded (meaning that there is a real number that is greater than the norm of every element in the Cauchy sequence).

Show that if $n \in \mathbb {N}$, then $\mathbf{F}^{n}$ is a Banach space with both the norms used in the first bullet point of Example 6.34.

Suppose $X$ is a nonempty set and $b(X)$ is the vector space of bounded functions from $X$ to $\mathbf{F}$. Prove that if $\left\| \cdot \right\|$ is defined on $b(X)$ by $\left\| f\right\| = \sup_{X}\left|f\right|$, then $b(X)$ is a Banach space.

Show that $\ell^{1}$ with the norm defined by $\left\| \left(a_{1}, a_{2}, \ldots \right)\right\|_{\infty } = \sup \left|a_{k}\right|$ is not a Banach space.

Show that $\ell^{1}$ with the norm defined by $\left\| \left(a_{1}, a_{2}, \ldots \right)\right\|_{1} = \sum_{k = 1}^{\infty }\left|a_{k}\right|$ is a Banach space.

Show that the vector space $C([0,1])$ of continuous functions from $[0,1]$ to $\mathbf{F}$ with the norm defined by $\left\| f\right\| = \int_{0}^{1}\left|f\right|$ is not a Banach space.

Suppose $U$ is a subspace of a normed vector space $V$ such that some open ball of $V$ is contained in $U$. Prove that $U = V$.

Prove that the only subsets of a normed vector space $V$ that are both open and closed are $\varnothing$ and $V$.

Suppose $V$ is a normed vector space. Prove that the closure of each subspace of $V$ is a subspace of $V$.

Suppose $U$ is a normed vector space. Let $d$ be the metric on $U$ defined by $d(f, g) = \left\| f-g\right\|$ for $f, g \in U$. Let $V$ be the complete metric space constructed in Exercise 16 in Section 6A.

Suppose $U$ is a subspace of a normed vector space $V$. Suppose also that $W$ is a Banach space and $S: U \rightarrow W$ is a bounded linear map.

For readers familiar with the quotient of a vector space and a subspace: Suppose $V$ is a normed vector space and $U$ is a subspace of $V$. Define $\left\| \cdot \right\|$ on $V / U$ by

Suppose $V$ and $W$ are normed vector spaces with $V \neq \left\{ 0\right\}$ and $T: V \rightarrow W$ is a linear map.

Suppose $U, V$, and $W$ are normed vector spaces and $T: U \rightarrow V$ and $S: V \rightarrow W$ are linear. Prove that $\left\| S \circ T\right\| \leq \left\| S\right\| \left\| T\right\|$.

Suppose $V$ and $W$ are normed vector spaces and $T: V \rightarrow W$ is a linear map. Prove that the following are equivalent:

1. $T$ is bounded.

2. There exists $f \in V$ such that $T$ is continuous at $f$.

3. $T$ is uniformly continuous (which means that for every $\varepsilon > 0$, there exists $\delta > 0$ such that $\left\| T f-T g\right\| < \varepsilon$ for all $f, g \in V$ with $\left\| f-g\right\| < \delta$).

4. $T^{-1}(B(0_W, r))$ is an open subset of $V$ for some $r > 0$.

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 $\overline{V \setminus U} = V$.

Suppose $U$ is a subset of a metric space $V$. Show that $U$ is dense in $V$ if and only if every nonempty open subset of $V$ contains at least one element of $U$.

Suppose $U$ is a subset of a metric space $V$. Show that $U$ has an empty interior if and only if $V \setminus U$ is dense in $V$.

Prove or give a counterexample: If $V$ is a metric space and $U, W$ are subsets of $V$, then $(\operatorname {int} U) \cup (\operatorname {int} W) = \operatorname {int}(U \cup W)$.

Prove or give a counterexample: If $V$ is a metric space and $U, W$ are subsets of $V$, then $(\operatorname {int} U) \cap (\operatorname {int} W) = \operatorname {int}(U \cap W)$.

Suppose

and $d(x, y) = \left|x-y\right|$ for $x, y \in X$.

Give an example of a metric space that is the countable union of closed subsets with empty interior.

Suppose $(X, d)$ is a complete metric space and $G_{1}, G_{2}, \ldots$ is a sequence of dense open subsets of $X$. Prove that $\bigcap_{k = 1}^{\infty } G_{k}$ is a dense subset of $X$.

Prove that there does not exist an infinite-dimensional Banach space with a countable basis.

Give an example of a Banach space $V$, a normed vector space $W$, a bounded linear map $T$ of $V$ onto $W$, and an open subset $G$ of $V$ such that $T(G)$ is not an open subset of $W$.