Functional Analysis_11. Weak Convergence_ 11.5 Weak-$*$ convergence

Consider $a=(a_k)_{k\in\mathbb{N}} \in \ell^\infty(\mathbb{R})$ and the sequence $\{x^{(n)}\}_{n\in\mathbb{N}} \subset \ell^\infty(\mathbb{R})$ defined by
$$x^{(n)} := (0, \dots, 0, a_{n+1}, a_{n+2}, \dots),$$
i.e., {\color{red} $x^{(n)}_k = a_{n+k}$ for all $k \in \mathbb{N}$}. Show that $x^{(n)} \xrightarrow{w^*} 0$. \vspace*{.4cm}
}

Clearly, $x^{(n)}_k = a_{n+k}$ for all $k \in \mathbb{N}$ gives
$$x^{(n)} = ( a_{n+1}, a_{n+2}, a_{n+3},\dots).$$

This must be a misprint: the displayed formula defines a cut-off tail, whereas the red “i.e.” defines a shifted tail; these are not the same sequence.

We will investigate both cases.

Видео Functional Analysis_11. Weak Convergence_ 11.5 Weak-$*$ convergence канала Dr. Szabó Tamás Zoltán