**Definition A..8** (Lipschitz continuous)
A function

is said to be Lipschitz
continuous with Lipschitz constant

, if

**Definition A..9** (Total Variation)
Let

be a decomposition of the interval

consisting of the
points

. If

is
function whose domain includes

, then

is called the variation of

with respect to

. If the upper
bound

of the set of all variations
with respect to all decompositions of

exists, then

is called the total variation of

over

. In
this case

is of bounded variation.

**Theorem A..10** (Weierstraß Approximation Theorem)
Let

be a real valued continuous function on the compact interval

. Then for each

there is a polynomial

in

indeterminates

such that

Put differently, the set of all polynomials on

is dense in the
set of the real valued continuous function on the compact
interval

.

The Stone-Weierstraß Theorem is a generalization of the Weierstraß
Approximation Theorem [47].

**Definition A..11**
A family of real valued functions

on

is said to separate the
points of

, if for two different points

and

of

there
is always an

such that

.

**Theorem A..12** (Stone-Weierstraß, Formulation 1)
Let

be a compact subset of a normed space

and

be a
subalgebra of

containing the function

and separating the
points of

. Then

is dense in

.

This is equivalent to the following formulation.

**Theorem A..13** (Stone-Weierstraß, Formulation 2)
Let

be a compact subset of a normed space

and

be a
closed subalgebra of

containing the function

and
separating the points of

. Then

.

**Theorem A..14** (Taylor)
Let

(

open) be a

function and

be
an

-neighborhood of

lying in

. Then for all

with

we have

where

**Theorem A..15** (Generalized Taylor Series)
Let

(

open) be an analytical function.
Then

holds.

Clemens Heitzinger
2003-05-08