Which of the following are Banach spaces?
A. The set of all real-valued functions $f$, $g$ which are functions of an independent real variable $t$ and are defined and continuous on the closed interval $[0,1]$, with norm
$$\|f\|=\max_{t \in [0,1]} |f(t)|. $$
B. The set of all continuous real-valued functions on $[0,1]$ and
$$\|f\|=\int_0^1 f(t) dt. $$
C. All polynomials on $(0,1)$ with complex coefficients with
$$\|f\| = \sup_{t \in [0,1]} |f(t)|. $$
My answer:
A. Yes, it is a Banach space as the set of all real-valued continuous functions on $[0,1]$ is complete with respect to the metric $$d(f,g)=\max_{t \in [0,1]} |f(t)-g(t)|.$$
Is it correct? I'm not able to complete parts B. and C..
Best Answer
A: Yes, your answer is correct. Perhaps you should prove it.
B: Imagine a function $f_n$ that is zero on $[0,1/2-1/n]$, $1$ on $[1/2 + 1/n, 1]$ and linear in between. Then compute its integral and its limit function (limit in the $L^1$-norm, assuming you are missing an absolute value in B).
C: Have you heard of the Stone Weierstrass approximation theorem? It tells you that with respect to the $\sup$-norm for any continuous function there is a polynomial arbitrarily close to it. Which means that for every continuous function there is a sequence of polynomials converging to it. Now let $f$ be any continuous function that is not a polynomial...