I got stuck with one, presumably, easy problem:
Let $\varphi: l^{\infty} \rightarrow l^{\infty}$ be linear operator given by $\varphi(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, \ldots) = (x_{1}, \frac{x_{2}}{2}, \frac{x_{3}}{3}, \ldots, \frac{x_{n}}{n}, \ldots)$.
From the given data, let us constuct an inverse limit for the sequence: $$ \newcommand{\la}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} l^{\infty} & \la{\varphi} & l^{\infty} & \la{\varphi} & l^{\infty} & \la{\varphi} & l^{\infty} & \la{\varphi} & l^{\infty} & \la{\varphi} & \ldots \end{array} $$ Is it possible to understand, which space the inverse limit of this sequence is topologically isomorphic to?
Remark: here so-called "inverse limit" indicates that for the given family $\{E_{\alpha} \}$ of topological vector spaces and the pair of indices $(\alpha, \beta)$ such that $\alpha \leq \beta$ there exists a continuous linear map $\psi_{\alpha \beta}: E_{\beta} \rightarrow E_{\alpha}$. So, the limit of such "inverse limit" is the subspace of a vector space $$\prod_{\alpha}{E_{\alpha}}$$ consisting of all elements $g \in \prod_{\alpha}{E_{\alpha}}$ such that $g(\alpha) = \psi_{\alpha \beta} g(\beta)$ for $\alpha \leq \beta$.