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## Main Question or Discussion Point

I am currently studying Stone–Weierstrass approximation theorem, and some parts really confuse me.

basically, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.

In the proof,

f(0)=f(1)=0.

g(x)= f(x)- f(0) - x[f(1)-f(0)] (0<= x <= 1)

Here g(0)=g(1)=0

My guess is that they want to define function f and then find a g(x) which looks like polynomials. Is that right? and also, is there any relationship between these transformation and uniformly convergence?

basically, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.

In the proof,

**they say on [a,b]=[0,1]**__at first__f(0)=f(1)=0.

__Why do they want to say that at the end points function must be zero?__**if the theorem is proved for this case, consider**__second,__g(x)= f(x)- f(0) - x[f(1)-f(0)] (0<= x <= 1)

Here g(0)=g(1)=0

__Can anyone explain the meaning and purpose of these transformation?__My guess is that they want to define function f and then find a g(x) which looks like polynomials. Is that right? and also, is there any relationship between these transformation and uniformly convergence?