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\begin{document}
\centerline{\bf Homework for MATH 4604 (Advanced Calculus II)}
\centerline{\bf Spring 2019}
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 13: Due on Tuesday 30 April}
\vskip.1in\noindent
13-1. Let \qquad $W\in\TNSR^+$, \qquad $\phi:\R\to W$.\\
Assume: \qquad $\,\forall k\in\scri_W$, \quad $0\in\dom[((\pi_k^W)\circ\phi)']$.\\
Show: \qquad $\,\,\,\,\,\,$ $0\in\dom[\phi']$.
\vskip.1in\noindent
13-2. Let \qquad $V,W\in\TNSR^+$, \qquad $f:V\drta W$, \qquad $q,u\in V$.\\
Assume: \qquad $\,\forall k\in\scri_W$, \quad $q\in\dom[\partial_u((\pi_k^W)\circ f)]$.\\
Show: \qquad $\,\,\,\,\,\,$ $q\in\dom[\partial_uf]$.
\vskip.1in\noindent
13-3. Let \qquad $V,W\in\TNSR^+$, \qquad $f:V\drta W$, \qquad $p\in V$.\\
Assume: \qquad $\,f''$ is continuous near $p$.\\
Show: \qquad $\,\,\,\,\,\,$ $\forall i,j\in\scri_V$, \quad $\partial_i\partial_jf$ is continuous near $p$.
\vskip.1in\noindent
13-4. Let  \qquad$V,W\in\TNSR^+$, \qquad $f:V\drta W$, \qquad $p\in V$.\\
Assume: \qquad $\,\,\forall h,i,j\in\scri_V$,\\
\phantom{x}\qquad\qquad\qquad\qquad $\partial_h\partial_i\partial_jf$ is both defined near $p$ and bounded near $p$.\\
Show: \qquad $\,\,\,\,\,\,$ $f''$ is continuous near $p$.
\vskip.1in\noindent
13-5. Let \qquad $V,W\in\TNSR^+$, \qquad $f:V\drta W$, \qquad $p\in V$.\\
Assume: \qquad $\,\,\forall h,i,j\in\scri_V$,\\
\phantom{x}\qquad\qquad\qquad\qquad $\partial_h\partial_i\partial_jf$ is both defined near $p$ and continuous at $p$.\\
Show: \qquad $\,\,\,\,\,\,$ $f'''$ is continuous at $p$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 12: Due on Tuesday 23 April}
\vskip.1in\noindent
12-1. Let $V,W\in\TNSR^+$, $f:V\drta W$, $k\in\N_0$.\\
Assume: \qquad $\forall m\in\scri_W$, \quad $(\pi_m^W)\circ f\in\scrlito_k^{V\R}$.\\
Show: \qquad $f\in\scrlito_k^{VW}$.
\vskip.1in\noindent
12-2. Let $V,W,X\in\TNSR^+$, $k\in\N_0$.\\
Show: \qquad $(\scrlito_k^{WX})\,\,\circ\,\,(\scrbigo_1^{VW})\quad\subseteq\quad\scrlito_k^{VX}$ \qquad and\\
\phantom{x}\qquad\qquad $\,\,\,(\scrbigo_1^{WX})\,\,\circ\,\,(\scrlito_k^{VW})\quad\subseteq\quad\scrlito_k^{VX}$ \qquad and\\
\phantom{x}\qquad\qquad $\,\,(\scrbigo_k^{WX})\,\,\circ\,\,(\scrbigo_1^{VW})\quad\subseteq\quad\scrbigo_k^{VX}$.
\vskip.1in\noindent
12-3. Let $V,W\in\TNSR^+$, $f:V\drta W$, $k\in\N_0$.\\
Assume: $f_{0_V}=0_W$.\\
Assume: $\forall j\in\scri_V$, $\partial_jf\in\scrbigo_k^{VW}$. \qquad\qquad
Show: $f\in\scrbigo_{k+1}^{VW}$.\\
\vskip.1in\noindent
12-4. Let $V,W\in\TNSR^+$, $f:V\drta W$, $k\in\N_0$.\\
Assume: $f_{0_V}=0_W$. \qquad\qquad
Assume: $\forall j\in\scri_V$, $(\partial_jf)_{0_V}=0_W$.\\
Assume: $\forall i,j\in\scri_V$, $\partial_i\partial_jf\in\scrbigo_k^{VW}$. \qquad\qquad
Show: $f\in\scrbigo_{k+2}^{VW}$.\\
\vskip.1in\noindent
12-5. Let $V,W\in\TNSR^+$, $f:V\drta W$.\\
Assume: $f_{0_V}=0_W$.\\
Assume: $\forall j\in\scri_V$, $\partial_jf$ is both defined near $0_V$ and continuous at $0_V$.\\
Show: $0_V\in\dom[f']$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 11: Due on Tuesday 16 April}
\vskip.1in\noindent
11-1. Let $\,$ $V:=\R^2$, $\,$ $S:=V\otimes V$, $\,$ $f:\R\drta V$, $\,$ $g:V\drta\R$.\\
Let $p\in\R$. \qquad $\,\,$
Assume \quad $f'_p=(1,2)$ \quad and \quad $f''_p=(3,4)$.\\
Let $q:=f_p$. \qquad
Assume \quad $g'_q=(5,6)$ \quad and \quad $\ds{g''_q=\left[\begin{array}{cc}7&8\\9&0\end{array}\right]}$.\\
\phantom{x}\qquad\qquad\qquad $\,\,\,\,\,$ Compute: \quad $(g\circ f)'_p$ \quad and \quad $(g\circ f)''_p$.
\vskip.1in\noindent
11-2. Let \quad $V:=\R^2$, \quad $S:=V\otimes V$, \quad $\ds{A:=\left[\begin{array}{cc}1&2\\2&4\end{array}\right]}$.\\
Show: \qquad\qquad $\,\,$ $\forall q\in V$, \quad $A\,\,\sbullet_S\,\,(q^{\otimes2})\,\,\ge\,\,0$.
\vskip.1in\noindent
11-3. Let $V:=\R^2$, $S:=V\otimes V$, $a,b,c\in\R$, $\ds{A:=\left[\begin{array}{cc}a&b\\b&c\end{array}\right]}$, $z:=0_2$.\\
Assume: \quad\qquad $\,\,$ $a>0$ \quad and \quad $ac-b^2>0$.\\
Show: \qquad\qquad $\,\,$ $\forall q\in V_z^\times$, \quad $A\,\,\sbullet_S\,\,(q^{\otimes2})\,\,>\,\,0$.
\vskip.1in\noindent
11-4. Let $\,$ $V:=\R^2$, $\,$ $S:=V\otimes V$, $\,$ $f:\R\drta V$, $\,$ $g:V\drta\R$.\\
Let $p\in\R$. \qquad $\,\,$
Assume \quad $f'_p\ne(0,0)$ \quad and \quad $f''_p\ne\frownie$.\\
Let $q:=f_p$. \qquad
Assume \quad $g'_q=(0,0)$ \quad and \quad $g''_q=\left[\begin{array}{cc}1&2\\2&5\end{array}\right]$.\\
Show: \qquad\qquad\qquad\qquad\qquad\quad $(g\circ f)''_p\,\,>\,\,0$.
\vskip.1in\noindent
11-5. Let $V,W\in\TNSR^+$, $f:V\drta W$, $i\in\scri_V$, $j\in\scri_W$, $W':=W\otimes V$.\\
Show: \qquad\qquad $(\pi_{j\|i}^{W'})\,\,\circ\,\,(f')\quad\subseteq\quad(\pi_j^W)\,\,\circ\,\,(\partial_if)$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 10: Due on Tuesday 9 April}
\vskip.1in\noindent
10-1. Let $V,W,X\in\TNSR^+$, \quad $f:V\drta W$, \quad $g:W\drta X$, \quad $u\in V$.\\
Show: \qquad $\forall p\in V$, \quad $(\partial_u^{VX}(g\circ f))_p\,\,\,\,=^*\,\,\,\,(g'_{f_p})\,\,*_{\R VW}\,\,((\partial_u^{VW}f)_p)$.
\vskip.1in\noindent
10-2. Let $S,V,W,Z\in\TNSR^+$, \quad $*\in\scrb_{VW}^Z$.\\
Let $f:S\drta V$, \quad $g:S\drta W$, \quad $u\in S$.\\
Show: \quad $\forall p\in S$, $\,\,$ $(\partial_u^{SZ}(f*g))_p\,\,=^*\,\,(\,((\partial^{SV}_uf)_p)*g_p\,)\,\,+\,\,(\,f_p*((\partial^{SW}_ug)_p)\,)$.
\vskip.1in\noindent
10-3. Let $V,W,X\in\TNSR^+$, \quad $A\in X\otimes V\otimes W$, \quad $y\in V$, \quad $z\in W$.\\
Show: \qquad $(A\,\,*_{\R,W,X\otimes V}\,\,z)\,\,*_{\R V\!X}\,\,y\quad=\quad A\,\,*_{\R,V\otimes W,X}\,\,(y\otimes z)$.
\vskip.1in\noindent
10-4. Let $V,W,X\in\TNSR^+$, $\,\,$ $f:V\drta W$, $\,\,$ $g:W\drta X$, $\,\,$ $u,v\in V$.\\
Show: \quad $\partial_v^{VX}\partial_u^{VX}(g\circ f)\,\,\,\,\supseteq\,\,\,\,(g''\circ f)\,\,\,\,*_{\R,W\otimes W,X}\,\,\,\,((\partial_u^{VW}f)\otimes(\partial_v^{VW}f))$\\
\phantom{x}\qquad\qquad\qquad\qquad\qquad\qquad $+$ \quad $(g'\circ f)\,\,\,\,*_{\R W\!X}\,\,\,\,(\partial_v^{VW}\partial_u^{VW}f)$.
\vskip.1in\noindent
10-5. Let $V:=\R^2$, \quad $W:=\R$, \quad $g:V\drta W$, \quad $z:=0_2$, \quad $i:=i_z^{\varepsilon_1^V}$.\\
Assume: $(g_z=0)\,\,\&\,\,((\partial_1^{VW}g)_z=0)$. \qquad
Show: $g\circ i\in\scrlito_1^{WW}$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 9: Due on Tuesday 2 April}
\vskip.1in\noindent
9-1. Let $V,W\in\TNSR^+$, \quad $\alpha:V\drta W$, \quad $\delta>0$, \quad $B:=B_V(0_V,\delta)$.\\
\phantom{xx}\quad Assume: $B\subseteq\dom[\alpha]$. \qquad\qquad
Assume: $\forall q\in B$, $|\alpha_q|_W\le|q|_V$.\\
\phantom{x}\qquad\qquad\qquad\qquad Show: $\alpha\in\scrbigo_1^{VW}$.
\vskip.1in\noindent
9-2. Let $S,T\in\TNSR^+$, \quad $f:S\drta T$, \quad $x,v\in S$.\\
\phantom{xx}\quad Show: \qquad$(\partial_v^{ST}f)_x\quad=^*\quad(f'_x)\,\,*_{\R ST}\,\,v$.
\vskip.1in\noindent
9-3. Let $V:=\R^2$, \quad $W:=\R$, \quad $f:V\drta W$, \quad $z:=0_2$.\\
\phantom{xx}\quad Assume: \qquad $(f'\hbox{ is defined near }z)\quad\&\quad(f_z=0)$.\\
\phantom{xx}\quad Let $g:=\partial_1^{VW}f$, \quad $h:=\partial_2^{VW}f$, \quad $i:=i_z^{\varepsilon_1^V}$.\\
\phantom{xx}\quad Show: $\exists\sigma\in\scrbigo_1^{WW}$, $\exists\tau\in\scrbigo_1^{VV}$ s.t.\\
\phantom{xx}\qquad\qquad $f\,\,=\,\,(g\circ i\circ\sigma\circ\pi_1^V)\cdot\pi_1^V\,\,+\,\,(h\circ\tau)\cdot\pi_2^V$ \qquad near $z$.
\vskip.1in\noindent
9-4. Let $V:=\R^2$, \quad $W:=\R$, \quad $h:V\drta W$, \quad $z:=0_2$.\\
\phantom{xx}\quad Assume: $z\in\dom[h']$. \quad
Assume: $h_z=(\partial_1^{VW}h)_z=(\partial_2^{VW}h)_z=0$.\\
\phantom{xx}\qquad\qquad\qquad\qquad Show: \qquad $h\in\scrlito_1^{VW}$.
\vskip.1in\noindent
9-5. Let $V:=\R^2$, \quad $W:=\R$, \quad $f:V\drta W$, \quad $z:=0_2$.\\
\phantom{xx}\quad Let $\alpha:=(\partial_1\partial_1f)_z$, \qquad $\beta:=(\partial_1\partial_2f)_z$, \qquad $\delta:=(\partial_2\partial_2f)_z$,\\
\phantom{xx}\qquad\qquad $\lambda:=(\partial_1f)_z$, \qquad $\mu:=(\partial_2f)_z$, \qquad $\rho:=f_z$.\\
Assume $\alpha\ne\frownie$, \quad $\beta\ne\frownie$, \quad $\delta\ne\frownie$, \quad $\lambda\ne\frownie$, \quad $\mu\ne\frownie$, \quad $\rho\ne\frownie$.\\
Define \quad $C\in\scrc_V^W$, \quad $L\in\scrl_V^W$, \quad $Q\in\scrq_V^W$ \quad by\\
\phantom{xx}\qquad\qquad $C(x,y)\,\,=\,\,\rho$, \qquad $L(x,y)\,\,=\,\,(\lambda,\mu)\,\,\sbullet_V\,\,(x,y)$,\\
\phantom{xx}\qquad\qquad\qquad $\ds{P(x,y)\,\,=\,\,\left[\begin{array}{cc}\alpha&\beta\\\beta&\delta\end{array}\right]\,\,\sbullet_{V\otimes V}\,\,((x,y)^{\otimes2})}$.\\
Let $\ds{R:=f-\left(C+L+\frac{P}{2!}\right)}$.\\
Show: $(\partial_1\partial_1R)_z=(\partial_1\partial_2R)_z=(\partial_2\partial_2R)_z=(\partial_1R)_z=(\partial_2R)_z=R_z=0$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 8: Due on Tuesday 26 March}
\vskip.1in\noindent
8-1. Let $S,T\in\TNSR^+$, \quad $R:S\drta T$.\\
\phantom{x}\qquad Assume $(\,\,R_{0_S}=0_T\,\,)\,\,\&\,\,(\,\,R'_{0_S}=0_{T\otimes S}\,\,)$. \qquad
Show: $R\in\scrlito_1^{ST}$.
\vskip.1in\noindent
8-2. Let $S,T\in\TNSR^+$, \quad $f:S\drta T$.\\
\phantom{x}\qquad Let $p\in\dom[f']$, \quad $L:=\Lin_{f'_p}^{ST}$, \quad $R:=f^T_p-L$.\\
\phantom{x}\qquad Show: $(\,\,R_{0_S}=0_T\,\,)\,\,\&\,\,(\,\,R'_{0_S}=0_{T\otimes S}\,\,)$.
\vskip.1in\noindent
8-3. Let $S\in\TNSR^+$, \quad $p,v\in S$, \quad $i:=i_p^v$. \qquad
Show: $i'=C_\R^v$.
\vskip.1in\noindent
8-4. Let $S,T\in\TNSR^+$, \quad $f:S\drta T$, \quad $p,v\in S$.\\
\phantom{x}\qquad Show: $(\partial_v^{ST}f)_p\,\,=^*\,\,(D_p^{ST}f)_v$.
\vskip.1in\noindent
8-5. Let $V:=\R^2$, \quad $W\in\R$, \quad $R:V\drta W$,\\
\phantom{x}\qquad\qquad\qquad\qquad\qquad $\delta>0$, \quad $J:=(-\delta;\delta)$, \quad $x\in J$.\\
\phantom{x}\qquad Assume: \quad $J^2\subseteq\dom[R']$.\\
\phantom{x}\qquad Show: \quad $\exists\alpha\in[0|x]$ \quad s.t. \quad $R|_{(0,0)}^{(x,0)}\,\,=\,\,((\partial_1^{VW}R)(\alpha,0))\cdot x$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 7: Due on Tuesday 12 March}
\vskip.1in\noindent
7-1. Let $V,W\in\TNSR$ and let $L,M\in\scrl_V^W$.\\
\phantom{x}\qquad\qquad\qquad Assume: $[L]_V^W=[M]_V^W$. \qquad\qquad
Show: $L=M$.
\vskip.1in\noindent
7-2. Let $V,W\in\TNSR$ and let $A\in W\otimes V$.\\
\phantom{x}\qquad\qquad\qquad Show: $\exists L\in\scrl_V^W$ s.t.~$[L]_V^W=A$.
\vskip.1in\noindent
7-3. Let $I$ be a finite set and let $z\in\R^I$.\\
\phantom{x}\qquad\qquad\qquad Show: \qquad $\ds{\sum_{j\in I}\,|z_j|\quad\le\quad\sqrt{\#\,I}\,\cdot\,\sqrt{\sum_{j\in I}\,z_j^2}}$.
\vskip.1in\noindent
7-4. Let $U,V,W\in\TNSR$ and let $B\in\scrb_{UV}^W$.\\
\phantom{x}\qquad\qquad\qquad Show: $\exists C\ge0$ s.t., $\forall p\in U$, $\forall q\in V$,\\
\phantom{x}\qquad\qquad\qquad\qquad\qquad\qquad $|B(p,q)|_W\quad\le\quad C\,\cdot|p|_U\,\,\cdot|q|_V$.
\vskip.1in\noindent
7-5. Let $T,U,V,W\in\TNSR$ and let $*\in\scrb_{UV}^W$.\\
\phantom{x}\qquad\qquad\qquad Show: $\BNZ_{TU}\,\,*\,\,\BNZ_{TV}\subseteq\BNZ_{TW}$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 6: Due on Tuesday 5 March}
\vskip.1in\noindent
6-1. Let $T\in\TNSR$, \quad $v\in T$. \qquad
Show: $\ds{v=\sum_{j\in\scri_T}\,v_j\varepsilon_j^T}$.
\vskip.1in\noindent
6-2. Let $S,T\in\TNSR$, \quad $A\in T\otimes S$, \quad $L:=\Lin_A^{ST}$, \quad $i\in\scri_S$.\\
\phantom{x}\qquad\qquad Show: $\ds{L(\varepsilon_i^S)\,=\,\sum_{j\in\scri_T}\,A_{j\|i}\,\cdot\,\varepsilon_j^T}$.
\vskip.1in\noindent
6-3. Let $S,T\in\TNSR^+$, \quad $C\in\scrc_S^T$, \quad $p\in S$. \qquad
Show: $D_p^{ST}C=\bfzero_S^T$.
\vskip.1in\noindent
6-4. Let $S,T\in\TNSR^+$, \quad $L\in\scrl_S^T$, \quad $p\in S$. \qquad
Show: $D_p^{ST}L=L$.
\vskip.1in\noindent
6-5. Let $V,W\in\TNSR^+$, \quad $Q\in\scrq_V^W$, \quad $B\in\scrs\scrb_V^W$.\\
\phantom{x}\qquad Assume: $Q=B(\bullet,\bullet)$. \qquad
Show: $\forall x\in V$, $D_x^{VW}Q=2\cdot(B(x,\bullet))$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 5: Due on Tuesday 26 February}
\vskip.1in\noindent
5-1. Let $S:=\R^2$ and let $T:=\R$. \quad
Show: $\scrq_{ST}\subseteq\scrbigo_2^{ST}$.
\vskip.1in\noindent
5-2. Let $S,T\in\TNSR^+$. \quad
Show: $(\BNZ_{ST})\cdot(\CVZ_{S\R})\subseteq\CVZ_{ST}$.
\vskip.1in\noindent
5-3. Let $S\in\TNSR^+$, \quad $f,g:\R\drta S$, \quad $p\in\R$.\\
\phantom{x}\qquad Show: $(f\sbullet g)_p^T\,\,=\,\,(f_p^T)\sbullet(g_p)\,+\,(f_p)\sbullet(g_p^T)\,+\,(f_p^T)\sbullet(g_p^T)$.
\vskip.1in\noindent
5-4. Let $S,T,U\in\TNSR^+$, \quad $f:S\drta T$, \quad $g:T\drta U$.\\
\phantom{x}\qquad Let $p\in S$, \quad $q:=f_p$. \qquad
Show: $(f\circ g)_p^T=(g_q^T)\circ(f_p^T)$
\vskip.1in\noindent
5-5. Let $S,T\in\TNSR$, \quad $i\in\scri_S$, \quad $j\in\scri_T$. \qquad\qquad
Show: $i\|j\in\scri_{S\otimes T}$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 4: Due on Tuesday 19 February}
\vskip.1in\noindent
4-1. Let $k\in\N_0$, $\alpha\in\scrbigo_k$ and $\ds{\lambda:=\adj_0^0\left(\frac{\alpha}{|\bullet|^k}\right)}$. \qquad
Show: $\lambda\in\BNZ$.
\vskip.1in\noindent
4-2. Let $f:\R\drta\R$ and let $U\in\R$.\\
Assume that $U$ is open in $\R$ and that $U\subseteq\dom[f']$.\\
Assume that $f$ is semi-increasing on $U$.\\
Let $T:=f'_*(U)$. \qquad
Show: $T\ge0$.
\vskip.1in\noindent
4-3. Let $f:\R\to\R$ and let $J$ be an interval.\\
Assume that $f$ is c/d on $J$. \qquad
Let $I:=\Int_\R J$ and $T:=f'_*(I)$.\\
Assume $T\ge0$. \qquad
Show: $f$ is semi-increasing on $J$.
\vskip.1in\noindent
4-4. Let $f:\R\drta\R$ and let $p\in\R$.\\
\phantom{xx}\quad Show: $(f(p+\bullet))'=f'(p+\bullet)$.
\vskip.1in\noindent
4-5. Let $f:\R\drta\R$ and let $p\in\dom[f]$.\\
Show: $[\,(\,f\hbox{ has a strict local minimum at }p\hbox{ in }\R\,)$\\
\phantom{x}\quad $\Leftrightarrow\,\,\,\,(\,f_p^T\hbox{ has a strict local minimum at }0\hbox{ in }\R\,)\,]$.
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\hrule
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\centerline{Homework 3: Due on Tuesday 12 February}
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3-1. Define $\alpha:\R\drta\R$ by $\ds{\alpha_h=\frac{h^2}{9\cdot(3+h)}}$. \qquad
Show: $\alpha\in\scrbigo_2$.
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3-2. Define $r:\R\drta\R$ by $r_x=1/x$. \qquad
Show: $r'_3=-1/9$.
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3-3. Define $L\in\scrl$ by $L_x=7x$. \qquad
Show: $L'=C_\R^7$.
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3-4. Let $a,b\in\R$. \qquad
Assume $a<b$. \qquad
Let $I:=(a;b)$, $J:=[a;b]$.\\
\phantom{x}\qquad\qquad Show: $\Int_\R J=I$ and $\Cl_\R I=J$.
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3-5. Find a continuous function $f:\R\to\R$ such that\\
\phantom{x}\qquad\qquad $[\,\,f(-2)=f(2)\,\,]\,\,\&\,\,[\,\,\forall x\in(-2;2),\,\,f'_x\ne0\,\,]$.
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\hrule
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\centerline{Homework 2: Due on Tuesday 5 February}
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2-1. Let $f:\R\drta\R$, $a\in\R$, $p\in\dom[f]$, $S\subseteq\R$ and $q:=f_p$.\\
Show: $(\,f>a\hbox{ on }S\,)\,\Leftrightarrow\,(\,f_p^T>a-q\hbox{ on }S-p\,)$.
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2-2. Show: $(\BNZ)\circ(\CVZ)\subseteq\BNZ$.
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2-3. Let $j,k\in\N$.
Show: $\scrbigo_j\circ\scrlito_k\subseteq\scrlito_{jk}$.
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2-4. Let $\phi\in\CVZ$, $\varepsilon>0$.
Show: $\exists B\in\scrb_\R(0)$ s.t.~$|\phi|<\varepsilon$ on $B$.
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2-5. Let $\alpha\in\scrlito_1$, $L\in\scrl\backslash\{\bfzero\}$.
Show: $\exists B\in\scrb_\R(0)$ s.t.~$|\alpha|\le|L|$ on $B$.
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\hrule
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\centerline{Homework 1: Due on Tuesday 29 January}
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1-1. Let $j\in\N_0$.
Show: $\scrlito_j\supseteq(\CVZ)\cdot(|\bullet|^j)$.
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1-2. Show: $\CVZ\subseteq\BNZ$.
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1-3. Show: $(\BNZ)\cdot(\BNZ)\subseteq\BNZ$.
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1-4. Show: $(\BNZ)\cdot(\CVZ)\subseteq\CVZ$.
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1-5. Let $j\in\N_0$.
Show: $\scrlito_j\subseteq\scrbigo_j$.
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\hrule
\end{document}

