Math Problems Vault

Problem 6B.1

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

\frac{1}{a+b i} = \frac{a}{a^{2}+b^{2}}-\frac{b}{a^{2}+b^{2}} i

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Problem 6B.2

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

\max \left\{ \left|\operatorname {Re} z\right|,\left|\operatorname {Im} z\right|\right\} \leq \left|z\right| \leq \sqrt{2} \max \left\{ \left|\operatorname {Re} z\right|,\left|\operatorname {Im} z\right|\right\}

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Problem 6B.3

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

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Problem 6B.4

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

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Problem 6B.5

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:

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

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Problem 6B.6

(a)(b)

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

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(a)

$f+g, f-g$ , and $f g$ are $\mathcal{S}$ -measurable functions;

(b)

if $g(x) \neq 0$ for all $x \in X$ , then $\frac{f}{g}$ is an $\mathcal{S}$ -measurable function.

Problem 6B.7

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

f(x) = \lim _{k \rightarrow \infty } f_{k}(x)

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

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Problem 6B.8

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

\int \alpha f d \mu = \alpha \int f d \mu

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Problem 6B.9

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

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Problem 6B.10

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

V \times W = \left\{ (f, g): f \in V \text{ and } g \in W\right\}

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

\left(f_{1}, g_{1}\right)+\left(f_{2}, g_{2}\right) = \left(f_{1}+f_{2}, g_{1}+g_{2}\right) \quad \text{ and } \quad \alpha (f, g) = (\alpha f, \alpha g)

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

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