James space

Example

Let $J$ be the set of real sequences $x = (x_n)_{n \ge 1}$ with $x_n \to 0$ and $\norm{x}_J < \infty$, where $$\norm{x}_J = \sup \Big( \sum_{i=1}^{k-1} (x_{p_i} - x_{p_{i+1}})^2 \Big)^{1/2}.$$ The supremum runs over all $k \ge 2$ and all integers $p_1 < p_2 < \cdots < p_k$. Then $\norm{\cdot}_J$ is a norm on $J$, and $J$ is a Banach space, the James space.

Check that $x_n = 1/\sqrt{n}$ lies in $J$. For $p_1 < \cdots < p_k$, the differences $a_i = x_{p_i} - x_{p_{i+1}}$ are positive. Then $\sum_i a_i^2 \le (\sum_i a_i)^2 = (x_{p_1} - x_{p_k})^2 \le 1$. So $\norm{x}_J \le 1$, although $x$ is not in $\ell^2$, since $\sum_n 1/n$ diverges.