maa6617-homework

MAA6617 Homework



  • Homework 11. Due Friday 13 April. Problem 25.25.
      Let $\mathcal A(\mathbb T)$ denote the set of function $f\in L^1(\mathbb T)$ such that the Fourier transform $\fhat$ belongs to $\ell^1(\mathbb Z)$.
      \begin{itemize}
      \item[a)] Prove, if $f,g\in \mathcal A(\mathbb T)$, then their product $fg$ also belongs to $\mathcal A(\mathbb T)$. (Hint: use the $\ell^1$ inversion theorem to write $f$ and $g$ as the sums of their Fourier series, and express the Fourier coefficients of $fg$ in terms of the coefficients of $f$ and $g$.) Thus, $\mathcal A(\mathbb T)$ is a ring.
      \item[b)] Prove the Fourier transform is a ring isomorphism from $\mathcal A(\mathbb T)$ {\em onto} $\ell^1(\mathbb Z)$ (where the multiplication on $\ell^1(\mathbb Z)$ is convolution).
      \end{itemize}

  • Homework 10. Due Wednesday 4 April. Problem 25.8 (in a version of the notes dated 26march2018 or later).
      This problem gives a proof that the Fourier transform $\widehat{\mbox{}}:L^1\to C_0(\mathbb R)$ is not surjective.
      \begin{itemize}
      \item[a)] Draw a picture of $h_n:=\one_{[-n,n]}* \one_{[-1,1]}$ and
      determine its $C_0(\mathbb R)$ norm.
      \item[b)] Show that $h_n$ is, up to a multiplicative constant independent of $n$, the
      Fourier transform of the $L^1$ function
      \begin{equation*}
      f_n := \frac{\sin 2\pi x \sin 2\pi nx}{x^2}.
      \end{equation*}
      (Hint: you can compute integrals, or use the $L^1$ inversion theorem.)
      \item[c)] Show that $\|f_n\|_1\to \infty$ as $n\to \infty$. Conclude that the Fourier transform is not surjective. (Hint: if it were surjective… .)

  • Homework 9. Due Wednesday 28 March. Problem 25.4.
      Prove, if $E\subset [0,1]$ has positive Lebesgue measure, then the set
      \begin{equation*}
      E-E =\{x-y:x,y\in E\}
      \end{equation*}
      contains an interval centered at the origin. (Hint: let $-E=\{-x:x\in E\}$ consider the function $h(x)=\one_{-E}*\one_E$ and apply Proposition 25.13.)

  • Homework 8. Due Monday 19 March. (Compare Problem 24.7).
      Let $(X,\mathscr{M},\mu)$ denote a measure space. If $\infty>p>r\ge 1$ and $L^r(\mu)\subset L^p(\mu)$, then $\mu$ does not admit sets of arbitrarily small positive measure; that is, there exists a $\delta >0$ such that if $E\in\mathscr{M}$ and
      $\mu(E)>0$, then $\mu(E)\ge \delta$. [Suggestion: Apply the closed graph theorem to the inclusion map $\imath: L^r\to L^p$. Here you may wish to use the result of Problem 24.9.] Conversely, if $\mu$ does not admit sets of arbitrarily small measure, and $1\le r\le p <\infty$, then $L^r(\mu)\subset L^p(\mu)$. As an example consider the $\ell^p$ spaces.

  • Homework 7. Due Monday 26 February. Problem 23.9.
      Suppose $H$ is a Hilbert space, $h\in H$ and $(h_n)$ is a sequence from $H$.
      \begin{itemize}
      \item[a)] Prove, if $(h_n)$ converges to $h$ in norm, then also $(h_n)$ converges to $h$ weakly. (Hint: Cauchy-Schwarz.)
      \item[b)] Prove, if $H$ is infinite-dimensional, and $(e_n)$ is an orthonormal sequence in $H$, then $e_n\to 0$ weakly, but $e_n\not\to 0$ in norm. (Thus weak convergence does not imply norm convergence.)
      \item[c)] Prove $(h_n)$ converges to $h$ in norm if and only if $(h_n)$ converges to $h$ weakly and $\|h_n\|\to \|h\|$.
      \item[d)] Prove if $(h_n)$ converges to $h$ weakly, then $\|h\|\le \liminf \|h_n\|$.
      \end{itemize}

  • Homework 6. Due Monday 19 February. Problem 23.2.
      Let $H$ be a Hilbert space and $T:H\to H$ a bounded linear operator.
      \begin{itemize}
      \item[a)] Prove there is a unique bounded operator $T^*:H\to H$ satisfying $\langle Tg,h\rangle = \langle g,T^*h\rangle$ for all $g,h\in H$, and $\|T^*\|=\|T\|$.
      \item[b)] Prove, if $S,T\in B(H)$, then $(aS+T)^* = \overline{a}S^*+T^*$ for all $a\in \mathbb K$, and that $T^{**}=T$.
      \item[c)] Prove $\|T^*T\| = \|T\|^2$.
      \item[d)] Prove $\text{ker}T$ is a closed subspace of $H$, $\overline{(\text{ran}T)} =(\text{ker}T^*)^\bot$ and $\text{ker}T^* = (\text{ran}T)^\bot$.
      \end{itemize}

  • Homework 5. Due Wednesday 14 February. Problem 22.10.
      \def\xx{\mathscr{X}}
      \def\mm{\mathscr{M}}
      \def\nn{\mathscr{N}}
      Suppose $\xx$ is a Banach space and $\mm$ and $\nn$ are closed subspaces. Show, if for each $x\in \xx$ there exist unique $m\in\mm$ and $n\in\nn$ such that
      \[
      x=m+n,
      \]
      then the mapping $P:\xx\to\mm$ defined by $Px=m$ is bounded.
      Suggestion: Consider the map $\mm \times \nn \to \xx$ defined by $(m,n)\mapsto x=m+n.$

  • Homework 4. Due Friday 9 February. Problem 22.6.
      \def\xx{\mathscr{X}}
      Suppose that $\xx$ is a vector space with a countably infinite basis. (That is, there is a linearly independent set $\{x_n\}\subset \xx$ such that every vector $x\in\xx$ is expressed uniquely as a {\em finite} linear combination of the $x_n$’s.) Prove there is no norm on $\xx$ under which it is complete. (Hint: consider the finite-dimensional subspaces $\xx_n:=\text{span}\{x_1,\dots x_n\}$.)

  • Homework 3. Due Monday 29 January.
      Let $B$ denote the subset of $\ell^\infty$ consisting of sequences which take values in $\{-1,1\}$. Show that any two (distinct) points of $B$ are a distance $2$ apart. Show, if $C$ is a countable subset of $\ell^\infty$, then there exists a $b\in B$ such that $\|b-c\|\ge 1$ for all $c\in C$. Conclude $\ell^\infty$ is not separable. Prove there is no isometric isomorphism $\Lambda : c_0\to\ell^\infty$. As a corollary, conclude that $c_0$ is not reflexive. (Of course, saying $c_0\ne \ell^\infty$ via the canonical embedding of $c_0$ into $c_{0}^{**}=\ell^\infty$ is much weaker than saying there is no isometric isomorphism between $c_0$ and $\ell^\infty$.)

  • Homework 2. Due Friday 26 January. Problem 21.4.
      Prove, if $\mathcal M$ is a {\em finite-dimensional} subspace of a Banach space $\mathcal X$, then there exists a closed subspace $\mathcal N \subset \mathcal X$ such that $\mathcal M \cap \mathcal N =\{0\}$ and $\mathcal M +\mathcal N=\mathcal X$. (In other words, every $x\in\mathcal X$ can be written uniquely as $x=y+z$ with $y\in \mathcal M$, $z\in\mathcal N$.) {\em Hint:} Choose a basis $x_1,\dots x_n$ for $\mathcal M$ and construct bounded linear functionals $f_1, \dots f_n$ on $\mathcal X$ such that $f_i(x_j)=\delta_{ij}$. Now let $\mathcal N=\cap_{i=1}^n \text{ker } f_i$.

  • Homework 1. Due Friday 12 January.
      \def\xx{\mathscr{X}}
      \def\yy{\mathscr{Y}}
      \def\emptynorm{\|\cdot\|}
      Suppose $\xx$ is a finite (say $n$) dimensional vector space. Prove all norms on $\xx$ are equivalent. Suggestion: Fix a basis $e_1, \dots e_n$ for $\xx$ and define $\|\sum a_k e_k\|_1 := \sum|a_k|$. It is routine to check that $\|\cdot\|_1$ is a norm on $\xx$. Now complete the
      following outline.
      \begin{enumerate}[(i)]
      \item Let $\|\cdot\|$ be the given norm on $\xx$. Show there is an $M$ such that $\|x\|\le M\|x\|_1$. Conclude that the mapping $\iota:(\xx,\emptynorm_1)\to (\xx,\emptynorm)$ defined by $\iota(x)=x$ is continuous;
      \item Show that the unit sphere $S=\{x\in\xx : \|x\|_1 =1\}$ in $(\xx,\emptynorm_1)$ is compact in the $\emptynorm_1$ topology;
      \item Show that the mapping $f:S\to \mathbb R$ given by $f(x)=\|x\|$ is continuous and hence attains its infimum. Show this infimum is not $0$ and finish the proof.
      \end{enumerate}

      Prove, if $\yy$ is a normed vector space, then every linear transformation $T:\xx\to\yy$ is bounded. Suggestion: First suppose the norm on $\xx$ is the norm $\emptynorm_1$ above.