Math Problems Vault

Problem 6C.1

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).

?

Problem 6C.2

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

\overline{B(f, r)} = \overline{B}(f, r).

?

Problem 6C.3

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

?

Remark.

?

Example 6.35 (second and third bullet points): for $n \in \mathbb {N}$ , define on $\mathbf{F}^{n}$

\left\| (a_1,\ldots ,a_n)\right\| = \left|a_1\right|^{1/2} + \cdots + \left|a_n\right|^{1/2}, \qquad \left\| (a_1,\ldots ,a_n)\right\| _{1/2} = \left(\left|a_1\right|^{1/2} + \cdots + \left|a_n\right|^{1/2}\right)^{2}.

$\left\| \cdot \right\|$ satisfies the positive definite condition and the triangle inequality but not the homogeneity condition; $\left\| \cdot \right\|_{1/2}$ satisfies the positive definite condition and the homogeneity condition but (for $n > 1$ ) not the triangle inequality.

Problem 6C.4

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).

?

Problem 6C.5

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.

?

Remark.

?

Example 6.34 (first bullet point): for $n \in \mathbb {N}$ , define $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{\infty }$ on $\mathbf{F}^{n}$ by

\left\| (a_1,\ldots ,a_n)\right\| _{1} = \left|a_1\right| + \cdots + \left|a_n\right|, \qquad \left\| (a_1,\ldots ,a_n)\right\| _{\infty } = \max \left\{ \left|a_1\right|,\ldots ,\left|a_n\right|\right\} .

Both $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{\infty }$ are norms on $\mathbf{F}^{n}$ .

Problem 6C.6

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.

?

Problem 6C.7

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.

?

Problem 6C.8

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.

?

Problem 6C.9

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.

?

Problem 6C.10

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

?

Problem 6C.11

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

?

Problem 6C.12

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

?

Problem 6C.13

(a)(b)(c)

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.

?

(a)

Show that the set $V$ is a vector space under natural operations of addition and scalar multiplication.

(b)

Show that there is a natural way to make $V$ into a normed vector space and that with this norm, $V$ is a Banach space.

(c)

Explain why (b) shows that every normed vector space is a subspace of some Banach space.

Problem 6C.14

(a)(b)(c)

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.

?

(a)

Prove that there exists a unique continuous function $T: \overline{U} \rightarrow W$ such that $\left.T\right|_{U} = S$

(b)

Prove that the function $T$ in part (a) is a bounded linear map from $\overline{U}$ to $W$ and $\left\| T\right\| = \left\| S\right\|$ .

(c)

Give an example to show that part (a) can fail if the assumption that $W$ is a Banach space is replaced by the assumption that $W$ is a normed vector space.

Problem 6C.15

(a)(b)(c)

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

\left\| f+U\right\| = \inf \left\{ \left\| f+g\right\| : g \in U\right\}

?

(a)

Prove that $\left\| \cdot \right\|$ is a norm on $V / U$ if and only if $U$ is a closed subspace of $V$ .

(b)

Prove that if $V$ is a Banach space and $U$ is a closed subspace of $V$ , then $V / U$ (with the norm defined above) is a Banach space.

(c)

Prove that if $U$ is a Banach space (with the norm it inherits from $V$ ) and $V / U$ is a Banach space (with the norm defined above), then $V$ is a Banach space.

Problem 6C.16

(a)(b)(c)(d)

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

?

(a)

Show that $\left\| T\right\| = \sup \left\{ \left\| T f\right\| : f \in V\text{ and }\left\| f\right\| < 1\right\}$ .

(b)

Show that $\left\| T\right\| = \sup \left\{ \left\| T f\right\| : f \in V\text{ and }\left\| f\right\| = 1\right\}$ .

(c)

Show that $\left\| T\right\| = \inf \left\{ c \in [0, \infty ):\left\| T f\right\| \leq c\left\| f\right\| \text{ for all }f \in V\right\}$ .

(d)

Show that $\left\| T\right\| = \sup \left\{ \frac{\left\| Tf\right\| }{\left\| f\right\| } \; : \; f \in V \text{ and } f \neq 0_V\right\}$ .

Problem 6C.17

?

Problem 6C.18

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

$T$ is bounded.
There exists $f \in V$ such that $T$ is continuous at $f$ .
$T$ 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 $\left\| f-g\right\| < \delta$ ).
$T^{-1}(B(0_W, r))$ is an open subset of $V$ for some $r > 0$ .

?