数学代写|MATH5011 Real Analysis

(#15 in Rudin) Under what conditions does equality hold in the Cauchy-Schwarz inequality? Recall that the aformentioned inequality says that if $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are complex numbers, then
$$
\left|\sum_{j=1}^n a_j \overline{b_j}\right|^2 \leq \sum_{j=1}^n\left|a_j\right|^2 \sum_{j=1}^n\left|b_j\right|^2 .
$$

Bonus Problem (#7 in Rudin) Fix $b>1, y>0$, and prove that there is a unique real $x$ such that $b^x=y$, by completing the following outline. (This is called the logarithm of $y$ to the base b.)
(a) For any positive integer $n, b^n-1 \geq n(b-1)$.
(b) Hence $b-1 \geq n\left(b^{1 / n}-1\right)$.
(c) If $t>1$ and $n>(b-1) /(t-1)$, then $b^{1 / n}y$, then $b^{w-(1 / n)}>y$ for sufficiently large $n$.
(f) Let $A$ be the set of all $w$ such that $b^w<y$, and show that $x=\sup A$ satisfies $b^x=y$.
(g) Prove that this $x$ is unique.

(a) Show that rational equivalence defines an equivalence relation on any subset of $\mathbb{R}$.
(b) Explicitly find a choice set for the rational equivalence relation on $\mathbb{Q}$.
(c) Define two numbers to be irrationally equivalent provided their difference is irrational. Is this an equivalence relation on $\mathbb{R} ?$ Is this an equivalence relation on $\mathbb{Q}$ ?

Show that any choice set for the rational equivalence relation on a set of positive outer measure must be uncountably infinite.

Let $E$ be a nonmeasurable set of finite outer measure. Show that there is a $G_\delta$ set $G$ that contains $E$ for which $m^(E)=m^(G)$ while $m^*(G \backslash E)>0$.