Function approximation

Remarks

Example: take $X = C[0, 1]$ with the supremum norm, $A$ the polynomials of degree at most $2$, and $f(x) = e^x$. The function $g(x) = 1 + x + x^2/2$ is an approximation within $0.22$. Indeed $e^x - g(x)$ is increasing on $[0, 1]$, since its derivative $e^x - 1 - x$ is nonnegative, so its maximum is $e - 2.5 < 0.22$.

Non-example: on $[-1, 1]$ with the supremum norm, the sign function cannot be approximated within any $\varepsilon < 1$ by continuous functions. Let $g$ be continuous with $g(0) = c$. Near $0$ the gaps $\abs{1 - g(x)}$ for $x > 0$ and $\abs{-1 - g(x)}$ for $x < 0$ come close to $\abs{1 - c}$ and $\abs{1 + c}$. One of these two numbers is at least $1$.

The choice of norm matters: the sign function can be approximated arbitrarily well by continuous functions in the $L^2$ norm.