This is a nitpick, but technically to apply the closed graph property instead of upper semicontinuity in Kakutani’s fixed point theorem, I believe you need to know that your LCTVS is metrizable. It is sufficient to show it has a countable local base by Theorem 9 of Rudin’s functional analysis, which is true because you’ve taken a countable power of R, but would not necessarily hold for any power of R (?).
This is a nitpick, but technically to apply the closed graph property instead of upper semicontinuity in Kakutani’s fixed point theorem, I believe you need to know that your LCTVS is metrizable. It is sufficient to show it has a countable local base by Theorem 9 of Rudin’s functional analysis, which is true because you’ve taken a countable power of R, but would not necessarily hold for any power of R (?).