Counter-example Stone-Weiertrass

Question

I want to find a counter-example for the Stone-Weiertrass theorem in the non-compact case. I am proposed to study the function $f(t) = \exp(t)$ on $\mathbb{R}$. Is $f$ uniform limit of polynomials? Is $f$ simple limit of polynomials? Thank you for your help!

Answer 1

You've come up with an excellent counterexample. Here's a hint in how to show that $f(t) = \exp(t)$ is not the uniform limit of polynomials: for a polynomial $p(t)$, note that $$ \lim_{n \to \infty}\left|\frac{p(t)}{\exp(t)}\right| = 0 $$ And that $$ \exp(t) - p(t) = \left[1- \frac{p(t)}{\exp(t)} \right]\exp(t) $$

Of course, $\exp(t)$ is the pointwise limit of polynomials as given by the partial sums of the Taylor series.

Answer 2

Suppose $e^x$ is the uniform limit of polynomials on $\mathbb {R}.$ Then there exists a polynomial $p$ such that $|p(x)-e^x| < 1,x \in \mathbb R.$ This implies $p$ is bounded on $(-\infty,0].$ But the only polynomials that are bounded on this interval are constant. Is there a constant $c$ such that $|e^x -c| < 1$ on $[0,\infty)?$ Of course not, so we have a contradiction.