臺灣大學數學系

八十八學年度第一學期碩博士班資格考試試題

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*任選五題,但3及10必選

1.
Suppose that $(\Omega ,\sigma ,\mu)$ is a measure space and $A_n \in \Sigma$, $n=1,2,3,\cdots$. Let E={ $x\in\Omega :x\in A_n$ for infinitely many $n$}.
(i) Show that E= \({\displaystyle{\bigcap_{k=1}^{\infty}\bigcup_{n\geq k}A_n}}\).
(ii) Show that μ(E)=0 if \({\displaystyle{\sum_n\mu (A_n)<+\infty}}\).
2.
Evaluate $\int_{-\infty}^{\infty}e^{-x^2}dx$ by showing that

\begin{displaymath}\left(\int_{-\infty}^{\infty}e^{-x^2}dx\right)^2=\int_{\mathbb{R}^2} e^{-(x^2+y^2)}dxdy =\int_0^{\infty}2\pi te^{-t^2}dt\end{displaymath}

3.
Let $\Omega = (0,\infty )\times (0,\infty )$ and $f(x,y)=e^{-xy},g(x,y) =e^{-xy}\sin x$ for $(x,y)\in\Omega$.
(i) Show that both $f$ and $g$ are not Lebesgue integrable on Ω.
(ii) From $\int_0^N[\int_0^Me^{-xy}\sin xdy]dx=\int_0^M[\int_0^Ne^{-xy} \sin xdx]dy$ for

$N>0,M>0$. Show that

\begin{displaymath}\int_0^N\frac{\sin x}{x}dx=\int_0^{\infty}\left[\int_0^Ne^{-xy} \sin xdx\right]dy\end{displaymath}

(iii) Following (ii) show that
\begin{displaymath}\int_0^{\infty}\frac{\sin x}{x}dx =\int_0^{\infty}\left[\int_0^{\infty}e^{-xy}\sin xdx\right]dy\end{displaymath}

(iv) Evaluate $\int_0^{\infty}e^{-xy}dx$ and show that $\int_0^{\infty}\frac{\sin x}{x}dx = \frac{\pi}{2}$.
4.
Suppose that $f$ is a Lebesgue integrable function on $\mathbb{R}$ and $\varphi\in C^1(\mathbb{R})$. Assume that both φ and $\varphi '$ are bounded functions. Define
\begin{displaymath}F(x)=\int_{-\infty}^{\infty}f(y)\varphi (x-y)dy\end{displaymath}

Show that $F\in C^1(\mathbb{R})$ and
\begin{displaymath}F'(x)=\int_{-\infty}^{\infty}f(y)\varphi '(x-y)dy\end{displaymath}

5.
State H$\ddot{o}$lder's inequality and use it to prove Minkowski's inequality.
6.
Let $K\subset C[0,1]$ be a family of absolutely continuous functions. Suppose that
1) \({\displaystyle{\sup_{f\in K} \vert f(0)\vert <\infty}}\),
2) $\int_0^1\vert f'(x)\vert^2dx\leq 1$ for all $f\in K$.
Show that $K$ is precompact in $C [0,1]$.
7.
Let $X$ be a Hilbert space and let $\{e_n\}_{n=1}^{\infty}$ be an orthonormal system in $X$. Consider $x=(x_i)_{j=1}^{\infty}\in l^2$ and for each $n$ let \({\displaystyle{V_n=\sum_{j=1}^{n}x_je_j}}\). Show that $\lim{n}{\infty}V_n$ exists in $X$ and $\vert\vert\lim{n}{\infty}V_n\vert\vert^2$ = \({\displaystyle{\sum_{j=1}^{\infty}\vert x_j\vert^2}}\).
8.
Prove that ``Monotone convergence Theorem'' for integrals is equivalent to ``Fatou's Lemma'' .
9.
Let $f$ be a function of bounded variation on [0,1].
(i) Show that $f$ is measurable.
(ii) Is it true that the total variation of $f$ is given by $\int_0^1\vert f'(x)\vert dx$ ?
10.
Let $\{f_n(z)\}_{n=1}^{\infty}$ be a sequence of analytic functions for $\vert z\vert<1$. Suppose that $f_n(z)$ converges uniformly to $f(z)$ on each compact subset of $\{z\in\mathbb{C}:\vert z\vert<1\}$. Show that if each $f_n$ is 1-1 , then so is $f$ unless $f$ is a constant function.


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