---- > [!definition] Definition. ([[uniform metric]]) > Given a set $X$ and a [[metric space]] $(Y,d)$, and given functions $f=(f(x))_{x \in X}$ and $g=(g(x))_{x \in X}$ in $Y^{X}$, we define a metric $\overline{\rho}$ on $Y^{X}$ by the equation > $ > \overline{\rho}(f,g):=\sup \{ \overline{d}(f(x), g(x)) : x \in X \}, > $ > where $\overline{d}$ denotes the [[standard bounded metric]] on $Y$. > We call the topology induced by $\overline{\rho}$ the **topology of uniform convergence** on $Y^{X}$ ([[uniform convergence#^equivalence-uniform-topology|see]]). > When the [[metric space|metric]] on $Y$ is [[metric induced by norm|induced]] by a [[norm]] $\|\cdot\|$, the uniform metric $\overline{\rho}$ on $Y^{X}$ is in turn induced by a [[norm]], namely the **sup norm** (also called **uniform norm**) $\|f\|_{\infty}:= \sup_{x \in X} \|f(x)\|.$ > [!basicexample] > Consider the [[uniform metric]] on the set $\mathbb{R}^{\mathbb{N}}$ of real sequences $(x_{n})_{n \geq 1}$, i.e., $d\big( (x_{n}), (y_{n}) \big)= \sup_{n \in \mathbb{N}} \{ \min (|x_{n} - y_{n}|, 1) \}.$Let $0 \leq r < 1$ and $x=(x_{n})_{n} \in \mathbb{R}^{\mathbb{N}}$. **(i).** Prove that $U(x,r):=(x_{1}-r, x_{1} + r) \times (x_{2}-r, x_{2}+r) \times \dots \times (x_{n}-r, x_{n}+r) \times \dots$ is *not* the ball of radius $r$ around $x$ (compare to [[euclidean, sup, and product topology are equal in finite dimensions|this result]]). > Set $y=(y_{n})_{n} = (x_{n} + r - \frac{1}{n})_{n \in \mathbb{N}}$. We see that $y_{1} \in (x_{1} -r, x_{1}+r),\dots, y_{n} \in (x_{n}-r, x_{n}+r), \dots, $ and so $y \in U(x,r)$, but $\begin{align} d(x,y) = & \sup_{n \in \mathbb{N}} \{ \min (\overbrace{|x_{n} - y_{n}|}^{\text{ always }<1}, 1) \} \\ = & \sup_{n \in \mathbb{N}} \{ |x_{n} - y_{n}| \} \\ = & \sup_{n \in \mathbb{N}} |x_{n} - \left( x_{n} + r - \frac{1}{n} \right)| \\ = & \sup_{n \in \mathbb{N}} |r - \frac{1}{n}| \\ \textgreater & r, \end{align}$ implying that $y$ is *not* in the ball of radius $r$ around $x$. > > **(ii).** Prove that $U(x,r)$ is in fact not open at all for the [[metric topology]] induced by $d$. > Take $y=(y_{n})_{n} = (x_{n} + r - \frac{1}{n})_{n \in \mathbb{N}}$ from above. Given $\varepsilon>0$, consider the $d$-open-ball $B_{d}(y, \varepsilon)$ of radius $\varepsilon$ around $y$. A point $p=(p_{n})_{n \in \mathbb{N}}$ belongs to $B_{d}(y,\varepsilon)$ if $d(p, y)= \sup_{n \in \mathbb{N}} \{ \min (|p_{n} - y_{n}|, 1 ) \} < \varepsilon.$ And $p$ belongs to $U(x,r)$ if $p_{1} \in (x_{1} -r, x_{1}+r),\dots, p_{n} \in (x_{n}-r, x_{n}+r), \dots. $ We claim that there exists $p \in \mathbb{R}^{\mathbb{N}}$ with $p \in B_{d}(y,\varepsilon)$ but $p \notin U(x,r)$. Indeed, set $p=(p_{n})_{n \in \mathbb{N}}:=\left( y_{n} + \frac{\varepsilon}{2} \right)_{n \in \mathbb{N}}$. Then, $p_{} \in B_{d}(y ,\varepsilon)$, for if $\varepsilon<2$ we have $\begin{align} d(p,y) = & \sup_{n \in \mathbb{N}} \{ \min (\overbrace{|p_{n} - y_{n}|}^{\text{ always }<1}, 1) \} \\ = & \sup_{n \in \mathbb{N}} \{ |p_{n} - y_{n}| \} \\ = & \sup_{n \in \mathbb{N}} \left\{ \frac{\varepsilon}{2} \right\} \\ = & \frac{\varepsilon}{2} \\ < & \varepsilon \end{align}$ and if $\varepsilon \geq 2$ we have $d(p,y) = \sup_{n \in \mathbb{N}}\min (\overbrace{|p_{n} - y_{n}|}^{\text{ always }\geq 1}, 1) = 1 < \varepsilon.$ However, using the [[archimedian property]] to fix $n$ large enough that $\frac{1}{n}< \frac{\varepsilon}{2}$, we see that $\begin{align} |p_{n} - x_{n}| = & |y_{n} + \frac{\varepsilon}{2} - x_{n}| \\ = & | x_{n} + r- \frac{1}{n} + \frac{\varepsilon}{2} - x_{n}| \\ = & |r + \underbrace{(- \frac{1}{n} + \frac{\varepsilon}{2}}_{>0}| \\ > & r, \end{align}$ implying that $p_{n} \notin (x_{n} -r, x_{n} + r)$ and hence $p \notin U(x,r)$. ^15c615 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```