Limits and Sequences of Functions

## The space $C\left(X\right)$

Let $X$ be a topological space. Let $ℳ X = f : X → ℂ$ be the algebra of complex valued functions on $X$.

The $*$ operation on $ℳ\left(X\right)$ is the map $*:ℳ\left(X\right)\to ℳ\left(X\right)$ given by

Let $X$ be a topological space. Let

The supremum norm on $C\left(X\right)$ is the function given by $f = sup x ∈ X f x .$

Define $d:C\left(X\right)×C\left(X\right)\to {ℝ}_{\ge 0}$ by $d f g = f - g .$

$C\left(X\right)$ is a complete metric space.

## Sequences of functions

Let $X$ be a topological space. Let $ℳ\left(X\right)$ be the algebra of complex valued functions on $X$. Let $\left({f}_{1},{f}_{2},\dots \right)$ be a sequence of functions in $ℳ\left(X\right)$.

The sequence $\left({f}_{1},{f}_{2},\dots \right)$ converges pointwise to $f:X\to ℂ$ if

The sequence $\left({f}_{1},{f}_{2},\dots \right)$ converges uniformly to $f:X\to ℂ$ if it is such that if $\epsilon \in {ℝ}_{\ge 0}$ then there exists $N\in {ℤ}_{>0}$ such that if $n\in {ℤ}_{>0}$ with $n\ge N$ then [???] I PERSONALLY FIND THIS MORE READABLE that is if $lim n → ∞ sup x ∈ X f n x - f x = 0 .$

Let $\left({f}_{1},{f}_{2},\dots \right)$ be a sequence of functions in $C\left(X\right)$. Then $\left({f}_{1},{f}_{2},\dots \right)$ converges in $C\left(X\right)$ if and only if $\left({f}_{1},{f}_{2},\dots \right)$ converges uniformly.

Let $X$ be a metric space and let $E\subseteq X$. Let $X$ be a point of $E$. Let $\left({f}_{1},{f}_{2},\dots \right)$ be a sequence of functions in $ℳ\left(E\right)$ and suppose that Then $lim n → ∞ lim t → x f n t = lim t → x lim n → ∞ f n t .$

## The Stone-Weierstrass theorem

If $f:\left[a,b\right]\to ℂ$ is a continuous function then there exists a sequence of polynomials $\left({p}_{1},{p}_{2},\dots \right)$ such that $\left({p}_{1},{p}_{2},\dots \right)$ converges uniformly to $f$.

Let $X$ be a metric space and let $E\subseteq X$. Let $𝒜$ [???] &Ascr; WORKING? be a subalgebra of $C\left(E\right)$.

The algebra $𝒜$ is self adjoint if it is such that if $f\in 𝒜$ then ${f}^{*}\in 𝒜$. The algebra separates points if it is such that if ${x}_{1},{x}_{2}\in E$ then there exists $f\in 𝒜$ such that $f\left({x}_{1}\right)\ne f\left({x}_{2}\right)$.

The algebra $𝒜$ vanishes at no point if it is such that if $x\in E$ then there exists $f\in 𝒜$ such that $f\left(x\right)\ne 0$.

Let $X$ be a metric space and let $K$ be a compact subset of $X$. Let $𝒜$ be a subalgebra of $C\left(K\right)$. If $𝒜$ is self adjoint, it separates points and it vanishes at no point of $K$ then $𝒜$ is dense in $C\left(K\right)$.