*1. Weierstrass Approximation Theorem*

We want to show that The set of polynomials is **dense** in the space of continuous functions. We shall firstly define the Bernstein Polynomials as

where *f* is a continuous function with domain [0,1] and we can thus see that it is uniformly continuous.This means that in δ-ε language :

here we manually set an interval in the domain. However, we are interested in the general relationship between two outputs without limitation ( other than the distance between two input must be in the range [0,1] ). So we want to now what happens when the |*x -ξ*| is larger than *δ*.

We recall the definition of norms in linear algebra. For finite vector space, the infinite norm is defined as:

It is easy to understand that this gives the largest entry (in terms of magnitude) of a vector. *Note that this is true even if there exists more than one maximum entry.* We expand this property into function space and define *M* to be the infinite norm of *f(x)*. Then for |*x -ξ*| is larger than *δ *we write:

Then combine (3) and (4) we obtain

We now want to use this relation to show that Bernstein polynomials can be used to approximate *f(x)*. We notice that the Bernstein polynomial of *f(ξ)* is just *f(ξ)*, where binomial expansion has been used to obtain this result*. *Then, using the fact that Bernstein polynomial is linear for *f(x),*

In the second line above we putted the first term of the function into the expression of the polynomial and it came out will some calculation. Obviously, we need to set *x=ξ *to proceed and this yields

What does this means? Remember that we have the freedom to make *n* as large as we want. This means that* *we can make the difference between Bernstein polynomials and our function as arbitrarily small as we want by approaching *n* to infinity. i.e., Bernstein polynomials converges to our *f(x)*.

[Literature: Matt Young, MATH 328 Notes, Queen’s University at Kingston, 2006]

*2. Taylor’s Theorem*

Now we are ready to prove Taylor’s Theorem. Knowing polynomials span the space of continuous function, we now assume that the function is also **infinitely differentiable**. In this special case, instead of using Bernstein polynomials, we want to use the basis consists of *1*,*x*,…,*x*^{n}. Matching the value of our series and our function at a particular point yields the ordinary form of single variable Taylor series.

*3. Vector fields Taylor Expansion*

To be continued…

Read *Taylor’s Theorem* in full