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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 6 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}


%%------------------------------------------------------------------------
\begin{exercise} % HW 11, Problem 1
    Recall that a function $f\colon V\to W$ between vector spaces is linear if $f(u+v)=f(u)+f(v)$ and $f(cv)=cf(v)$ hold, for all $u,v\in V$ and $c\in F$. Alternatively, it is linear if
    \[
    f(au+bv)=af(u)+bf(v),\qquad\text{for all $u,v\in V$ and $a,b\in F$}.
    \]
    Prove that these two definitions are equivalent.
\end{exercise}

%\begin{solution} 
    % To do. Recall that you need to prove both $\Rightarrow$ and $\Leftarrow$ inmplications separately. 
%\end{solution}

\smallskip

%%------------------------------------------------------------------------
\begin{exercise} % HW 11, Problem 2
    Let $f\colon V\to W$ be a linear function between vector spaces.
    \begin{enumerate}[label=(\alph*)]
        \item Prove that $f(\bm{0})=\bm{0}$, and that $f(-v)=-f(v)$, for all $v\in V$. 
        \item The \emph{nullspace} of $f$ is the set
        \[
        \NS(f)=\big\{v\in V:f(v)=\bm{0}\big\}.
        \]
        Prove that $\NS(f)$ is a subspace of $V$.
    \end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do. You just need to show that $\NS(f)$ is closed under addition and scalar multiplication.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 11, Problem 3
Prove that the open interval $(a,b)$ in the standard metric $(\R,d)$ is an open set. 
\end{exercise}

%\begin{solution} 
    % To do. Use the definition; this is very short.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 11, Problem 4
    Let $A,B\subseteq X$ be open sets in a metric space $(X,d)$. 
    Prove that $A\cup B$ and $A\cap B$ are also open.
\end{exercise}

%\begin{solution} 
    % To do. Use the definition; this is very short.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 11, Problem 5
Let $X=\mathcal{C}([0,1])$ be the set of continuous functions on $[0,1]$. Consider the two metric spaces, $(X,d_1)$ and $(X,d_\infty)$, where
\[
d_1(f,g)=\int_0^1\big|f(x)-g(x)\big|\,dx,\qquad\qquad
d_\infty(f,g)=\max_{x\in[0,1]}\big|f(x)-g(x)\big|.
\]
Now, define the sequence of functions $(f_n)_{n\geq 1}$, where
\[
f_n(x)=\begin{cases} 1-nx & 0\leq x<\tfrac{1}{n}, \\ 0 & \tfrac{1}{n}<x\leq 1.\end{cases}
\]
Picture of the graphs of $f_1,\dots,f_4$ are shown below. 
\[
\begin{tikzpicture}[scale=2,>=Stealth]  
  \begin{scope}[shift={(0,0)}]
  \draw[->] (-.25,0) -- (1.25,0) node[right] {$x$};
  \draw[->] (0,-.25) -- (0,1.25) node[above] {$y$};
  \draw (1,-.05)--(1,.05);  \draw (-.05,1)--(.05,1);
  \node at (-.2,1) {\footnotesize $1$};
  \node at (1,-.2) {\footnotesize $1$};
  %
  \draw[very thick] (0,1)--(1,0);
  \node at (.65,.8) {\small $f_1(x)$};
  \end{scope}
  %%
  \begin{scope}[shift={(2,0)}]
  \draw[->] (-.25,0) -- (1.25,0) node[right] {$x$};
  \draw[->] (0,-.25) -- (0,1.25) node[above] {$y$};
  \draw (1,-.05)--(1,.05);  \draw (-.05,1)--(.05,1);
  \node at (-.2,1) {\footnotesize $1$};
  \node at (1,-.2) {\footnotesize $1$};
  %
  \draw[very thick] (0,1)--(.33,0)--(1,0);
  \node at (.65,.8) {\small $f_2(x)$};
  \end{scope}
  %%
  \begin{scope}[shift={(4,0)}]
  \draw[->] (-.25,0) -- (1.25,0) node[right] {$x$};
  \draw[->] (0,-.25) -- (0,1.25) node[above] {$y$};
  \draw (1,-.05)--(1,.05);  \draw (-.05,1)--(.05,1);
  \node at (-.2,1) {\footnotesize $1$};
  \node at (1,-.2) {\footnotesize $1$};
  %
  \draw[very thick] (0,1)--(.33,0)--(1,0);
  \node at (.65,.8) {\small $f_3(x)$};
  \end{scope}
  %%
  \begin{scope}[shift={(6,0)}]
  \draw[->] (-.25,0) -- (1.25,0) node[right] {$x$};
  \draw[->] (0,-.25) -- (0,1.25) node[above] {$y$};
  \draw (1,-.05)--(1,.05);  \draw (-.05,1)--(.05,1);
  \node at (-.2,1) {\footnotesize $1$};
  \node at (1,-.2) {\footnotesize $1$};
  %
  \draw[very thick] (0,1)--(.25,0)--(1,0);
  \node at (.65,.8) {\small $f_4(x)$};
  \end{scope}
\end{tikzpicture}
\]
Answer the following questions for both metric spaces, $(X,d_1)$ and $(X,d_\infty)$.
\begin{enumerate}[label=(\alph*)]
    \item Compute $d(f_n,0)$, where $0$ is the zero function.
    \item Characterize the functions in the unit ball $B_1(0)$ in terms of their graphs. 
    \item Determine whether the sequence $(f_n)$ converges or diverges. Prove your answer.
\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 11, Problem 6
Define the \emph{comb metric} on $X=\R^2$ by the distance function
\[
d_{\text{comb}}(x,y)=\begin{cases}|y_1-y_2| & \text{if $x_1=x_2$}, \\
|y_1|+|x_1-x_2|+|y_2| & \text{if $x_1\neq x_2$}.
\end{cases}
\]
Describe and sketch each of the following open balls in $(X,d_{\text{comb}})$.
\begin{multicols}{3}
\begin{enumerate}
    \item $B_1\big((0,0)\big)$
    \item $B_1\big((0,\tfrac{1}{2})\big)$
    \item $B_1\big((0,1)\big)$.
\end{enumerate}
\end{multicols}
\end{exercise}

%\begin{solution} 
    % To do. Use TikZ for the sketches, and modify the figure I
    % provided in the text.
%\end{solution}
\medskip

%%------------------------------------------------------------------------

\end{document}