On page 184 of Reed-Simon, there is an example differentiating the notions of convergence in the sequence space $\ell^2$. In particular it makes the following statements
Consider the following families of maps $T_n, S_n, W_n: \ell^2 \to \ell^2$.
- Define $T_n: (x_1, x_2, \dots) \mapsto (n^{-1} x_1, n^{-1} x_2, \dots)$. Then $T_n \to 0$ uniformly.
- Define $S_n: (x_1, x_2, \dots) \mapsto (\underbrace{0, \dots, 0}_{n \text{ terms}}, x_{n+1}, x_{n+2}, \dots)$. Then $S_n \to 0$ strongly but not uniformly.
- Define $W_n: (x_1, x_2, \dots) \mapsto (\underbrace{0, \dots, 0}_{n \text{ terms}}, x_1, x_2, \dots)$. Then $W_n \to 0$ weakly but not strongly or uniformly.
There are no justifications given, but are the following justifications correct?
- Let $x \in \ell^2$. Then $\|T_n x\|_{\ell^2} = n^{-1} \|x\|_{\ell^2}$, so $\|T_n - 0 \|_{{\ell^2}^\star} = \|T_n \|_{{\ell^2}^\star} = n^{-1} \to 0$.
- Let $x \in \ell^2$. Since $\|x\|_{\ell^2} < \infty$, then $\sqrt{\sum_{k > n} x_k^2} \underset{n \to \infty}{\to} 0$. Then $\|S_n x - 0x\|_{\ell^2} = \sqrt{\sum_{k > n} x_k^2} \underset{n \to \infty}{\to} 0$. To see that $S_n \not \to 0$ uniformly, observe that $\|S_n - 0\|_{(\ell^2)^\star} \geq \|S_n e_{n+1}\|_{\ell^2} = 1$, for all $n$.
- Let $x \in \ell^2$. Since $(\ell^2)^\star = \ell^2$, to prove that $W_n \to 0$ weakly, it suffices to show that for any $y \in \ell^2$, $|\sum_{n+1}^{\infty} y_i x_{i - n} | \to 0$. Let $N = \|x\|_{\ell^2}$. Since $\|y_n\|_{\ell^2} < \infty$, $|\sum_{n+1}^{\infty} y_i x_{i - n} | \leq (\sum_{n+1}^\infty y_i^2)^{1/2}N \to 0$ (we applied the triangle inequality and then Cauchy Schwarz). To see that $W_n \not \to 0$ strongly, note that $\|W_n e_1\|_{\ell^2} = \|e_{n+1}\|_{\ell^2} = 1$ for all $n$. Since uniform convergence implies strong convergence, this shows $W_n \not \to 0$ uniformly as well.