Problem 4
Define $$ e^{\theta i}=\cos \theta+i \sin \theta $$ Describe \(e^{\theta i}\) geometrically. Is \(\left|e^{\theta i}\right|=1 ?\)
Problem 4
A set \(A\) in \((S, \rho)\) is said to be totally bounded iff for every \(\varepsilon>0\) (no matter how small \(), A\) is contained in a finite union of globes of radius \varepsilon. By Problem 3 , any such set is bounded. Disprove the converse by a counterexample. [Hint: Take an infinite set in a discrete space.]
Problem 4
Find the edge-lengths of \(A=(\bar{a}, \bar{b})\) in \(E^{4}\) if $$ \bar{a}=(1,-2,4,0) \text { and } \bar{b}=(2,0,5,3) $$ Is \(A\) a cube? Find some rational points in it. Find \(d A\) and \(v A\).
Problem 5
Metrize the extended real number system \(E^{*}\) by $$ \rho^{\prime}(x, y)=|f(x)-f(y)|, $$ where the function $$ f: E^{*} \underset{\text { onto }}{\longrightarrow}[-1,1] $$ is defined by \(f(x)=\frac{x}{1+|x|}\) if \(x\) is finite, \(f(-\infty)=-1,\) and \(f(+\infty)=1\) Compute \(\rho^{\prime}(0,+\infty), \rho^{\prime}(0,-\infty), \rho^{\prime}(-\infty,+\infty), \rho^{\prime}(0,1), \rho^{\prime}(1,2),\) and \(\rho^{\prime}(n,+\infty)\). Describe \(G_{0}(1), G_{+\infty}(1),\) and \(G_{-\infty}\left(\frac{1}{2}\right)\). Verify the metric axioms (also when infinities are involved).
Problem 5
Take for granted the lemma that $$ a^{1 / p} b^{1 / q} \leq \frac{a}{p}+\frac{b}{q} $$ if \(a, b, p, q \in E^{1}\) with \(a, b \geq 0\) and \(p, q>0,\) and $$ \frac{1}{p}+\frac{1}{q}=1 $$ (A proof will be suggested in Chapter \(5, \S 6,\) Problem 11.) Use it to prove Hölder's inequality, namely, if \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1,\) then $$ \sum_{k=1}^{n}\left|x_{k} y_{k}\right| \leq\left(\sum_{k=1}^{n}\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}\left|y_{k}\right|^{q}\right)^{\frac{1}{q}} \text { for any } x_{k}, y_{k} \in C . $$
Problem 5
Compute (a) \(\frac{1+2 i}{3-i}\) (b) \((1+2 i)(3-i) ;\) and (c) \(\frac{x+1+i}{x+1-i}, x \in E^{1}\). Do it in two ways: (i) using definitions only and the notation \((x, y)\) for \(x+y i\); and (ii) using all laws valid in a field.
Problem 6
Give examples of incomplete metric spaces possessing complete subspaces.
Problem 6
Prove (for \(E^{2}\) and \(E^{3}\) ) that $$ \bar{x} \cdot \bar{y}=|\bar{x}||\bar{y}| \cos \alpha $$ where \(\alpha\) is the angle between the vectors \(\overrightarrow{0 x}\) and \(\overrightarrow{0 y}\); we denote \(\alpha\) by \(\langle\bar{x}, \bar{y}\rangle .\)
Problem 7
Prove that the perpendicular distance of a point \(\bar{p}\) to a plane \(\vec{u} \cdot \bar{x}=c\) in \(E^{n}\) is $$ \rho\left(\bar{p}, \bar{x}_{0}\right)=\frac{|\vec{u} \cdot \bar{p}-c|}{|\vec{u}|} . $$ \(\left(\bar{x}_{0}\right.\) is the orthogonal projection of \(\bar{p},\) i.e., the point on the plane such that \(\left.\overrightarrow{p x_{0}} \| \vec{u} .\right)\)
Problem 7
Let $$ \begin{aligned} z &=r(\cos \theta+i \sin \theta) \\ z^{\prime} &=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right), \text { and } \\ z^{\prime \prime} &=r^{\prime \prime}\left(\cos \theta^{\prime \prime}+i \sin \theta^{\prime \prime}\right) \end{aligned} $$ as in Corollary 2. Prove that \(z=z^{\prime} z^{\prime \prime}\) if $$ r=|z|=r^{\prime} r^{\prime \prime}, \text { i.e., }\left|z^{\prime} z^{\prime \prime}\right|=\left|z^{\prime}\right|\left|z^{\prime \prime}\right|, \text { and } \theta=\theta^{\prime}+\theta^{\prime \prime} . $$ Discuss the following statement: To multiply \(z^{\prime}\) by \(z^{\prime \prime}\) means to rotate \(\overrightarrow{0 z^{\prime}}\) counterclockwise by the angle \(\theta^{\prime \prime}\) and to multiply it by the scalar \(r^{\prime \prime}=\) \(\left|z^{\prime \prime}\right| .\) Consider the cases \(z^{\prime \prime}=i\) and \(z^{\prime \prime}=-1 .\)
