A.2 The Distributive Law For Cross Product
Here is a proof of the distributive law for the cross product. Let x be a vector. From the above
for all x. In particular, this holds for x = a×
a × b
+ a × c
and this proves the distributive law for the cross product.
Weierstrass Approximation Theorem
An arbitrary continuous function defined on an interval can be approximated uniformly by a polynomial,
there exists a similar theorem which is just a generalization of this which will hold for continuous functions
defined on a box or more generally a closed and bounded set. However, we will settle for the case of a box
first. The proof is based on the following lemma.
Lemma B.0.1 The following estimate holds for x ∈
and m ≥
Proof: First of all, from the binomial theorem
Take a derivative and then let t = 1.
Take another time derivative of both sides.
Plug in t
Then it follows
and from what was just shown, this equals
Thus the expression is maximized when x = 1∕2 and yields m∕4 in this case. This proves the lemma.
With this preparation, here is the first version of the Weierstrass approximation theorem.
Theorem B.0.2 Let f ∈ C
Then these polynomials converge uniformly to f on
denote the largest value of
. By uniform continuity of
, there exists a δ >
By the binomial theorem,
is large enough. ■
Corollary B.0.3 If f ∈ C
, then there exists a sequence of polynomials which converge
uniformly to f on
Proof: Let l :
be one to one, linear and onto. Then
is continuous on
and so if
0 is given, there exists a polynomial p
such that for all x ∈
Therefore, letting y = l
it follows that for all y ∈
As another corollary, here is the version which will be used in Stone’s generalization later.
Corollary B.0.4 Let f be a continuous function defined on
. Then there
exists a sequence of polynomials
such that pm
Proof: From Corollary B.0.3 there exists a sequence of polynomials
Simply consider pm