$\newcommand{\br}{\\}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\P}{\mathbb{P}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\L}{\mathcal{L}}$ $\newcommand{\spa}[1]{\text{span}(#1)}$ $\newcommand{\dist}[1]{\text{dist}(#1)}$ $\newcommand{\max}[1]{\text{max}(#1)}$ $\newcommand{\min}[1]{\text{min}(#1)}$ $\newcommand{\supr}[0]{\text{sup}}$ $\newcommand{\infi}[0]{\text{inf}}$ $\newcommand{\ite}[1]{\text{int}(#1)}$ $\newcommand{\ext}[1]{\text{ext}(#1)}$ $\newcommand{\bdry}[1]{\partial #1}$ $\newcommand{\argmax}[1]{\underset{#1}{\text{argmax }}}$ $\newcommand{\argmin}[1]{\underset{#1}{\text{argmin }}}$ $\newcommand{\set}[1]{\left\{#1\right\}}$ $\newcommand{\emptyset}{\varnothing}$ $\newcommand{\tilde}{\text{~}}$ $\newcommand{\otherwise}{\text{ otherwise }}$ $\newcommand{\if}{\text{ if }}$ $\newcommand{\proj}{\text{proj}}$ $\newcommand{\union}{\cup}$ $\newcommand{\intercept}{\cap}$ $\newcommand{\abs}[1]{\left| #1 \right|}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$ $\newcommand{\pare}[1]{\left(#1\right)}$ $\newcommand{\brac}[1]{\left[#1\right]}$ $\newcommand{\t}[1]{\text{ #1 }}$ $\newcommand{\head}{\text H}$ $\newcommand{\tail}{\text T}$ $\newcommand{\d}{\text d}$ $\newcommand{\limu}[2]{\underset{#1 \to #2}\lim}$ $\newcommand{\limd}[3]{\underset{#1 \to #2; #3}\lim}$ $\newcommand{\der}[2]{\frac{\d #1}{\d #2}}$ $\newcommand{\derw}[2]{\frac{\d #1^2}{\d^2 #2}}$ $\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\pderw}[2]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\pderws}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}$ $\newcommand{\inv}[1]{{#1}^{-1}}$ $\newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $\newcommand{\nullity}[1]{\text{nullity}(#1)}$ $\newcommand{\rank}[1]{\text{rank }#1}$ $\newcommand{\nullspace}[1]{\mathcal{N}\pare{#1}}$ $\newcommand{\range}[1]{\mathcal{R}\pare{#1}}$ $\newcommand{\var}[1]{\text{var}\pare{#1}}$ $\newcommand{\cov}[2]{\text{cov}(#1, #2)}$ $\newcommand{\tr}[1]{\text{tr}(#1)}$ $\newcommand{\oto}{\text{ one-to-one }}$ $\newcommand{\ot}{\text{ onto }}$ $\newcommand{\ceil}[1]{\lceil#1\rceil}$ $\newcommand{\floor}[1]{\lfloor#1\rfloor}$ $\newcommand{\Re}[1]{\text{Re}(#1)}$ $\newcommand{\Im}[1]{\text{Im}(#1)}$ $\newcommand{\dom}[1]{\text{dom}(#1)}$ $\newcommand{\fnext}[1]{\overset{\sim}{#1}}$ $\newcommand{\transpose}[1]{{#1}^{\text{T}}}$ $\newcommand{\b}[1]{\boldsymbol{#1}}$ $\newcommand{\None}[1]{}$ $\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$ $\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$ $\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$ $\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$ $\newcommand{\vcw}[2]{\begin{matrix} #1 \br #2 \end{matrix}}$ $\newcommand{\vce}[3]{\begin{matrix} #1 \br #2 \br #3 \end{matrix}}$ $\newcommand{\vcr}[4]{\begin{matrix} #1 \br #2 \br #3 \br #4 \end{matrix}}$ $\newcommand{\vct}[5]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \end{matrix}}$ $\newcommand{\vcy}[6]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{matrix}}$ $\newcommand{\vcu}[7]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{matrix}}$ $\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$ $\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$ $\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$ $\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$ $\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$ $\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$ $\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$ $\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$ $\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$ $\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$ $\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$ $\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$
Note: Different Types of Limits
  1. (limits of sequences of points in a metric space)
    $\limu{ n }{ \infty } x^{(n)} $
  2. (limiting values of functions at a point)
    $\limd{ x }{ x_0 }{ x \in E } f(x) $
  3. (pointwise limits of sequences of functions)
    $f^{(n)}$
  4. (uniform limits of functions)
    $f^{(n)} $

Notice that (4) can be viewed as a special case of (1).

Definition: Metric Spaces of Bounded Functions

Suppose $(X, d_X) $ and $(Y, d_Y) $ are metric spaces. We let $B(X \to Y) $ denote the space of bounded functions from $X$ to $Y$:

$$B(X \to Y) \triangleq \set{ f | f: X \to Y\text{ is a bounded function}} $$

We define a metric $d_ \infty : B(X \to Y) \times B(X \to Y) \to \R^+$ as

$$d_ \infty (f, g) \triangleq \supr_{x \in X}{ d_Y(f(x), g(x))}$$

for all $f, g \in B(X \to Y)$.

The metric is known as the sup norm metric or the $L^ \infty $ metric.

We will also use $d_{B(X \to Y)} $ as a synonym for $d_ \infty $.

Note: $\None{}d\_\infty(f,g)$ Is Always Finite

The distance $d_\infty(f,g)$ is always finite in $B(X \to Y) $.

Proof 
  </span>
</span>
<span class="proof__expand"><a>[expand]</a></span>

$f, g \in B(X \to Y)$

Then,

$\exists y_0, y_1 \in Y, r_1, r_2 \in \R, \forall x \in X,$

$$f(x) \in B(y_0, r_1), g(x) \in B(y_1, r_2)$$

That is

$$d_Y(f(x), y_0) < r_1, d_Y(g(x), y_1) < r_2$$

Then

$$\begin{align*} d_Y(f(x), g(x)) &\leq d_Y(f(x), y_0) + d_Y(y_0, y_1) + d_Y(y_1, g(x)) \br &= r_1 + r_2 + d_Y(y_1, y_2) < \infty \end{align*}$$

Hence

$$d_ \infty(f,g) = \supr_{x \in X}{ d_Y(f(x), g(x))} < \infty$$

Proposition

$(X, d_X), (Y, d_Y)$. Let $(f^{(n)})_{n =1}^ \infty \subseteq B(X \to Y) $, and let $f$ be another function in $B(X \to Y)$. Then,

$(f^{(n)})^\infty_{ n =1}$ converges to $f$ in the metric $d_{B(X \to Y)}$ if and only if $(f^{(n)})^\infty_{n =1}$ converges uniformly to $f$.

Theorem : The Space of Continuous Functions Is Complete

$(X, d_X), (Y, d_Y)$. $(Y, d_Y)$ complete. The space $(C(X \to Y), d_{B(X \to Y)}|_{C(X \to Y) \times (X \to Y)})$ is a complete subspace of $(B(X \to Y), d_{B(X \to Y)}) $. In other words, every Cauchy sequence of functions in $C(X \to Y) $ converges to a function in $C(X \to Y) $.