Eufisky - The lost book


  1. Let $f_n\in L^2(R)$ be a sequence of measurable functions over the line, $f_n\rightarrow f$ almost everywhere. Let $||f_n||_{L^2}\rightarrow||f||_{L^2}$, prove that $||f_n-f||_{L^2}\rightarrow 0$.

    enlightenedProof.(Weingarten) $L^2(\mathbf R)$ is a Hilbert space, and $\|f_n\|_{L^2}\to\|f\|_{L^2}$ as $n\to\infty$ , so we only need to prove $f_n\in f$ weakly in $L^2(\mathbf R)$, that is, for each $g\in L^2(\mathbf R)$, there holds

    \[\lim_{n\to\infty}\int_{\mathbf R}f_ng=\int_{\mathbf R} fg.\]

    For each $\epsilon>0$, there exists $R>0$, such that \[\int_{|x|>R}|g|^2<\epsilon^2,\]

    and by the absolute continuity of integration of $g$, there exists a positive $\delta$, such that: for any Lebesgue measurable subset $E$ of $\mathbf R$ with $m(E)<\delta$, there holds \[\int_E|g|^2<\epsilon^2.\]

    By the Egoroff's thoerem, there exists a subset $E_\delta$ of $(-R,R)$ with $m((-R,R)\setminus E_\delta)<\delta$, such that the convergence $\lim\limits_{n\to\infty}f_n=f$ is uniform on the $E_\delta$, so there exists $N\in\mathbf Z_+$, such that \[(\forall n>N,x\in E_\delta),(|f_n-f|<\epsilon/\sqrt{2R}).\]

    Assume $M=\|g\|_{L^2}+\|f\|_{L^2}+\sup\limits_n\|f_n\|_{L^2}$, hence $\forall n>N$, we have the following estimations

    \begin{align*}\int_{\mathbf R}|f_n-f|\cdot|g|&=\left(\int_{E_\delta}+\int_{(-R,R)\setminus E_\delta}+\int_{|x|>R}\right)|f_n-f|\cdot|g|\\&\leq\sqrt{\int_{E_\delta}|f_n-f|^2\cdot\int_{E_\delta}|g|^2}+\sqrt{\int_{(-R,R)\setminus E_\delta}|f_n-f|^2\cdot\int_{(-R,R)\setminus E_\delta}|g|^2}\\&+\sqrt{\int_{|x|>R}|f_n-f|^2\cdot\int_{|x|>R}|g|^2}\\&\leq M\epsilon+2\sqrt{2}M\epsilon.\end{align*}

    the proof is finished.


  2. Let $f$ be a continuous function on $[a,b]$, define $M_n=\int ^b_a f(x)x^n{\rm d}x$. Suppose that $M_n=0$ for all $n$, show that $f(x)=0$ for all $x$.
  3. Determine all entire functions $f$ that satisfying the inequality $$|f(z)|\leq|z|^2|{\rm Im}(z)|^2$$ for $z$ sufficlently large.
  4. Describe all holomorphic functions over the unit disk $D=\{z||z|\leq 1\}$ which maps the boundary of the disk into the boundary of the disk.
  5. Let $T:H_1\rightarrow H_2, Q:H_2\rightarrow H_1$ be bounded linear operators of Hilbert spaces $H_1,\ H_2$. Let $QT={\rm Id}-S_1,TQ={\rm Id}-S_2$ where $S_1$ and $S_2$ are compact operators. Prove ${\rm Ker}T=\{v\in H_1,Tv=0\},{\rm Coker}T=H_2/\overline{{\rm Im}T}$, where ${\rm Im}T=\{Tv\in H_2,v\in H_1\}$ are finite dimensional and ${\rm Im}(T)$ is closed in $H_2$.
  6. Let $H_1$ be the Sobolev space on the unit interval $[0,1]$, i.e. the Hilbert space consisting of functions $f\in L^2([0,1])$ such that $$||f||_1^2=\sum^\infty_{n=-\infty}(1+n^2)|\hat f(n)|^2<\infty;$$ where $$\hat f(n)=\frac 1 {2\pi}\int_0^1f(x)e^{-2\pi inx}{\rm d}x$$ are Fourier coefficients of $f$. Show that there exists constant $C>0$ such that $$||f||_{L^\infty}\leq C||f||_1$$ for all $f\in H_1$, where $||\cdot||_{L^\infty}$ stands for the usual supremum norm. (Hint: Use Fourier series.)