Equicontinuity and boundedness
Exercise 5.1
For each $k \in \mathbb{N}$, let $f_k : \ell^2 \to \mathbb{R}$ be given by $f_k(\{x_n\}_{n=1}^\infty) = x_k$. Is the family $F = \{f_k\}_{k\in\mathbb{N}}$ equicontinuous? Is it uniformly bounded? Is it pointwise bounded?
The Weierstrass approximation theorem
Exercise 5.2
Suppose that $f \in C([0, 1])$ and that $\int_0^1 f(x) x^n \, dx = 0$ for all nonnegative integers $n$. Prove that $f$ must be identically zero.