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\homework{3: Vector Spaces and Metrics}{Due Friday, Week 2}

Do \textbf{one} of the \textbf{three} problems below!   Prove all of your claims.  
\begin{enumerate}
\item \begin{enumerate}
\item Let $\mathcal{C}(\mathbb{R})$ denote the collection of all continuous functions on the real numbers.  Is this a vector space over $\mathbb{R}$?   Prove your answer.  
\item Let $\not\mathcal{C}(\mathbb{R}) \cup \{0\}$ denote the collection of all discontinuous functions on the real numbers, along with the identically-0 function $f(x) = 0$. Is this a vector space over $\mathbb{R}$?   Prove your answer.  
\end{enumerate}
\item Given a field $\langle F, +, \cdot \rangle$, we say that a subset $G \subseteq F$ is a \textbf{subfield} of $F$ if $\langle G, +, \cdot \rangle$ is a field in its own right (if we interpret $+, \cdot$ here as the same operations that we used on $F$.)  For example, $\mathbb{Q}$ is a subfield of $\mathbb{R}$, and $\mathbb{R}$ is a subfield of $\mathbb{C}$.  Conversely, $\mathbb{Z}/2\mathbb{Z}$ is not a subfield of $\mathbb{Q}$; even though 0 and 1 are both elements of $\mathbb{Q}$, the addition operation on $\mathbb{Z}/2\mathbb{Z}$ does not agree with the addition operation on $\mathbb{Q}$.
\begin{enumerate}
\item Find a field $\langle F, +, \cdot \rangle$ whose only subfield is itself.
\item Show that if $\langle F, +, \cdot \rangle$ is a field and $G$ is a subfield of $F$, then $F$ is a \textbf{vector space} over the field $G$.
\item Show that not all vector spaces have this structure: i.e. find a vector space $V$ over a field $F$ such that $F$ is not a subfield of $V$.
\end{enumerate}
\item Consider the following metric, called the ``post office'' metric, on $\mathbb{R}^n$:
\begin{itemize}
\item For any $\vec{x} \in \mathbb{R}^n$, we have $d(\vec{x}, \vec{x}) = 0$.
\item For any $\vec{x} \neq \vec{y} \in \mathbb{R}^n$, we have $d(\vec{x}, \vec{y}) = ||\vec{x}|| + ||\vec{y}||$.  (If you haven't seen this notation before, $||\vec{x}||$ denotes the quantity $\sqrt{x_1^2 + \ldots x_n^2}$.)
\end{itemize}
The idea with this metric is that if you want to mail a package from some location $\vec{x}$ to another location $\vec{y}$, you often have to route it through some central packing location (in this case, $\vec{0}$.)

Prove that the post office metric is a metric.
\end{enumerate}


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