Limits and Sequences of Functions [???] PAGE TITLE

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 23 October 2009

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

Let $X$ be a topological space. Let $$ℳ\left(X\right)=\left\{f:X\to ℂ\right\}$$ 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 $$f^{*}\left(x\right)=\overline{f\left(x\right)}\text{,}\text{for }x\in X\text{.}$$

Let $X$ be a topological space. Let $$C\left(X\right)=\left\{f:X\to ℝ|f\text{ is continuous and bounded}\right\}\text{.}$$

The supremum norm on $C\left(X\right)$ is the function $‖ ‖:C\left(X\right)\to ℝ$ given by $$‖f‖=\underset{x\in X}{\sup }\left|f\left(x\right)\right|\text{.}$$

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

$C\left(X\right)$ 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 $$f\left(x\right)=\underset{n\to \infty }{\lim }{f}_n\left(x\right)\text{,}\text{for }x\in X.$$

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 $N\in {\mathbb{Z}}_{>0}$ such that if $n\in {\mathbb{Z}}_{>0}$ with $n\ge N$ then $$\left|f_n\left(x\right)-f\left(x\right)\right|\le \epsilon \text{,}\text{for all }x\in X,$$ [???] I PERSONALLY FIND THIS MORE READABLE that is if $$\underset{n\to \infty }{\lim }\left(\underset{x\in X}{\sup }\left|f_n\left(x\right)-f\left(x\right)\right|\right)=0\text{.}$$

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 $$\underset{t\to x}{\lim }{f}_n\left(t\right)\text{exists for each }n\in {\mathbb{Z}}_{>0}\text{.}$$ Then $$\underset{n\to \infty }{\lim }\underset{t\to x}{\lim }{f}_n\left(t\right)=\underset{t\to x}{\lim }\underset{n\to \infty }{\lim }{f}_n\left(t\right)\text{.}$$

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 $𝒜$ [???] 𝒜 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 $C\left(K\right)$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)