Solution
= Solution
\b[(iv) implies (iii):] $P_k=\sum_{n=1}^kx_n\varphi_n$ is a finite sum of bounded <rank>-one operators.
\b[(iv) implies (v):] continuity gives $\varphi_n(y_r)\to0$ whenever $y_r\to0$, so any scalar limit $\beta$ must vanish.
\b[(iv) implies (vi):] fix $n$. Continuity gives $\varphi_n(y_r)\to\varphi_n(x)$, while the assumed uniform coordinate convergence gives $\varphi_n(y_r)\to\beta_n$. Uniqueness of the scalar limit yields $\beta_n=\varphi_n(x)$ for every $n$.