The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval with endpoints $a$ and $b$, carrying the distance $d_{\mathbb{R}}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv), let $(Y,d_{Y})$ be a metric space, and let $w:[a,b]\to\mathbb{R}$ be a map continuous on $[a,b]$ with $w(t)>0$ for every $t\in[a,b]$. Continuity of a map $[a,b]\to Y$ refers to $d_{\mathbb{R}}$ on $[a,b]$ and $d_{Y}$ on $Y$, as in Continuous Map Between Metric Spaces (ii).

(i) $C([a,b],Y)$ is the set of maps $x:[a,b]\to Y$ continuous on $[a,b]$, the subset

$$C([a,b],Y)=\{x\in Y^{[a,b]}:x\text{ is continuous on }[a,b]\}$$

of the set $Y^{[a,b]}$ of maps from $[a,b]$ to $Y$, which is a set by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii); its condition, that $x$ is continuous at $s$ relative to $[a,b]$ for every $s\in[a,b]$, is expressed by a predicative formula by A Predicative Formula for Continuity at a Point, and the Set of Real-Valued Maps Continuous at a Point (i). By The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i), a set $x$ lies in $C([a,b],Y)$ if and only if $x$ is a map $x:[a,b]\to Y$ continuous on $[a,b]$.

(ii) Let $x,y\in C([a,b],Y)$. The weighted uniform distance of $x$ and $y$ is

$$d_{w}(x,y)=\max_{t\in[a,b]}w(t)\,d_{Y}(x(t),y(t)),$$

the greatest element of the set $\{w(t)\,d_{Y}(x(t),y(t)):t\in[a,b]\}\subseteq\mathbb{R}$, unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iii). It exists: the map $h:[a,b]\to\mathbb{R}$, $t\mapsto d_{Y}(x(t),y(t))$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)), is continuous on $[a,b]$ by The Distance Between the Values of Two Continuous Maps Is Continuous (i), applied with $(\mathbb{R},d_{\mathbb{R}})$ in place of $(X,d_{X})$ and $[a,b]$ in place of $A$. Let $g:[a,b]\to\mathbb{R}$ be the product $t\mapsto w(t)h(t)$. It is continuous on $[a,b]$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v), and Extreme Value Theorem on a Closed Real Interval (ii) gives $q\in[a,b]$ with $g(t)\le g(q)$ for every $t\in[a,b]$, so that $g(q)$ is a greatest element of that set. $d_{w}$ also denotes the map $C([a,b],Y)\times C([a,b],Y)\to\mathbb{R}$, $(x,y)\mapsto d_{w}(x,y)$, given by an expression as in Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices (iv).

(iii) For all $x,y\in C([a,b],Y)$, the maximum in (ii), unfolded by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iii), is an element of the set $\{w(t)\,d_{Y}(x(t),y(t)):t\in[a,b]\}$ that bounds it above; that is, $w(t)\,d_{Y}(x(t),y(t))\le d_{w}(x,y)$ for every $t\in[a,b]$, and there is $t_{0}\in[a,b]$ with $w(t_{0})\,d_{Y}(x(t_{0}),y(t_{0}))=d_{w}(x,y)$.